I've always liked the Turtledove books as an entertaining fiction, with the character of Clarence Potter being a very interesting and at times entertaining character. But as said Turtledove goes completely off the rails historically speaking after 1910 to completely outlandish proportions, but it's still an interesting read.
Also I gotta throw some stuff Turtledove got wrong weapons wise, majorly wrong, I might add. His putting Lee-Enfield SMLE's in the hands of the Confederate Army through WW1 and beyond as a "Tredegar Rifle" is an interesting and in all probability possible thing, in regards to copying British standards, (we'll get into that later), but after that it gets murky and completely impossible. I'll number it and go down the list of wrongs and "most likely's".
1. M1903 Springfield- The M1903 of this timeline would most likely be way different than in ours, as in our timeline the M1903 was "inspired" with some American ideas thrown in by the Spanish M1893 Mauser and Germany's development of the Gewehr 98 after the Mauser designs showed itself superior to the US Krag-Jorgensen in the Spanish-American War. Since in this timeline there was no Spanish-American War, how would such inspiration have come about? Probably not at all. Since in Turtledove's universe the USA and German Empire are essentially allies, and the US had been in a slump so to speak, its probably more likely an American M1903 of this era would most likely be a US licensed copy of Germany's Gewehr 98 of WW1 fame in our timeline.
2. John Moses Browning- I'm lumping a lot guns together in this one. John Browning was probably responsible for more firearms designs and contributions to firearms development than anyone else in to this day. Its impossible to overestimate his contributions. The Colt M1911, the B.A.R., the M1917 Water-Cooled Machine gun, the M1886 M1892 M1894 and M1895 Winchester lever actions, the very automatic pistol operating system in America along with many, many other gun in America and Europe, sprang from his mind. However in Turtledove's timeline, he probably never would have designed a single gun. John Browning was a Mormon, (sometimes Mormon missionary) and Utah native, and in this timeline with so much bloodshed in Utah, along with it becoming a police state in Utah at the time he started working as a gunsmith for his dad, and with gun ownership apparently an underground thing, and the first Utah rebellion happening before he designed his first gun, its highly likely he never would have been allowed to design or build a gun. Even if he had, chances are he would never sell them because the US, a power oppressing his people, would get them, so in all probability none of them would have ever existed. Plus with his contributions to firearm development in Europe, there's a chance guns of the WW1 and WW2 eras would have been more backward without him.
3. The .45 Auto cartridge- This cartridge, the one the famous M1911 was chambered for, was like the M1903 Springfield developed in response to the Spanish-American War, with on of the designers being John Thompson. Without that war, and especially without John Browning to design a gun to fire it, it never would have been thought of probably till at least after WW1.
4. The Thompson SMG- John Thompson got his start in the Spanish-American War with the then new machine gun. He along with John Parker took a battery of Gatling guns against orders and supported Roosevelt's charge up San Juan Hill. That experience, was instrumental in the forming of the philosophy of a hand held machine gun to Thompson, along with WW1 showing him the best type of gun to go for, a "trench broom". Without those experiences, its more than likely to think the famous Thompson SMG never would have thought of.
5. The "Tredegar Automatic Rifle"- Yeah this never would have been a thing. Descriptions of this gun fit an our timeline US M14, and evolution of the famous M1 Garand. Such a design never would have been possible, for first off, Garand was a Canadian, and the Turtledove timeline version of Canada would preclude a Canadian from designing the M1, so an M14 spawn of it never would have happened. Another thing, Garand, was a student of John Browning at one point, so definitely never would have happened without the Garand gas system inspired by John Browning.
6. CS SMG's- From what I recall, most descriptions of them fit in line with the Soviet PPSH-41, PPS-43, and US M3 Grease Gun. It highly improbable the two Soviet SMG's could have been designed without the copying of ideas from John Browning, so that's a nope, the US M3, it may have been possible to exist for it though, but its a big if.
7. SMG designs- In a timeline without John Browning, all subguns most likely would have been based off the German designs. The MP18 of our timeline may have come into existence, along with the Italian designs that evolved from their odd ball gun from WW1. Seeing how the WW2 British Sten was on offshoot of the Lanchester offshoot of the German MP28, along with Italy not being a global power per se, plus them being on the losing end of WW1, along with Britain, I think US and CS SMG's would more likely be offshoots of the German and British copies of German designs.
8. Automatic pistols in general- Without the influence of John Browning, its safe to say if the US decided on adopting a German design would have been likely. BUT would they have existed? Maybe, maybe not. The first automatic pistol was designed by Hugo Borchardt in the early 1890's, from Europe, after he had migrated to the US, worked as a designer for Sharps, and then went back. It's not improbable to say without the conditions of our timeline, this may have never happened. So automatics may have went an entirely different route, the first gun I know of to come out after Borchardt would be the Mauser C1896. Another thing to remember is the famous German Luger was a result of the 1893 Borchardt, and may have never existed without it. Would it have happened without Borchardt and his American experience? Who knows...
9. US decision to adopt a big bore handgun cartridge- This decision came out of a small bore wave that swept the civilized world after the invention of smokeless powder. The US like many countries dropped it smoke belching .45 Colt and .45 S&W cartridges and adopted the .38 Colt in the M1889 Colt New Army and the New Navy. The Spanish-American War's results changed that philosophy. The Philippines was brought under US "influence" after that little war and shortly afterwards the Philippine Insurrection happened. Muslim Moro warriors got drugged up and the small .38 Colt cartridge didn't do anything to stop them in close quarters. However some re-arsenaled Colt M1873's had been issued before hand when enough newer guns weren't available, and the big .45 dropped them like a bad habit. Without that experience, its safe to say the US never would have went back to big bullets.
10 The Lee-Enfield family of guns- In Turtledove's timeline, this gun was the main gun of the Confederate Army in WW1 and used into WW2, and in ours was the British standard in different versions from the late 1880's into the late 1940's. But with Confederate Independence, however it may have never existed. James Paris Lee was a child of Scottish immigrants to America and was trained as a watchmaker. During the CW he turned his attention to designing guns, with his first design being marketed to the US Army in 1865, (no purchases from what I can recall). But if the US lost the War in 1862, he may have never gone into gun design, and therefore it never could have been adopted by the Brits and sold to the Confederates in any version. This is as catastrophic to gun design in general as is the loss of John Browning, because, Lee invented the first practical detachable magazine for rifles. No James Paris Lee gun designer, no detachable magazine in any gun for a long, long time.
In conclusion, without John Browning, and without the seemingly inconsequential Spanish-American War, practically nothing we know of from our timeline would have existed. Guns, cartridges, philosophies none of it. Its safe to say throughout WW1 its more likely the USA would have been running around with knockoffs of German guns like the Mausers, and the CSA with straight knockoffs of British guns, with maybe some Confederate copies of older US designs, like the Colt M1873. The way the post "War of Secession" USA is spoken of, its even possible the US Army never would have adopted instrumental guns like the Colt M1873 due to costs, and the "Second Mexican War" was too early to effect philosophy's into WW1 with some of the mentioned characters, as it was before the some of the great designers got into the business or the invention of the biggest gamechanger in modern warfare, smokeless gunpowder. There's so many things implausible in that department, when the facts are its more likely armies would have been marching into battle with revolvers and bolt-action rifles as their standard weapons into the 1970's, changing tactics and philosophy irreparably in any alternate timeline. Heck without such important people in gun design and bad circumstance, uniforms and gear could have resembled WW1 and late 19th Century stuff for a long time too. Heck the philosophy of general issue camo uniforms, or darker earthier toned uniforms may have been put off for a long while.
Still a very entertaining read though. Also please forgive the length of this post, even with a brief synopsis I put forward not in depth like this, there's still a ton of ground to cover.