I know that is a Post-1880 Oxs & Bucks badge. But I do have a question about the designation "1/43" as this is a little foreign to us Yanks.
I first came across the British designations while studying WW2 organization. You would find "8 Royal Fusiliers" which we interpret as 8th Regiment. But it is actually the 8th battalion of the Royal Fusilier Regiment. They learned after WW1 to assign troops from one region into other theaters of war. Then sometimes you come across some really strange designations: " 1/5 West Surrey" or " 2/5 Leicestershire". I think I got that.
Q: What does "1/43" designate in the Napoleonic Wars? A battalion? Or smaller?
Okay, so here's the way it works for infantry up until 1881.
Each Regiment of Foot has its own distinct existence - for example in this case the
43rd was the 43rd (Monmouthshire).
This regiment is administrative (and usually had a "depot" back home of a couple of companies that trained up replacements). Each regiment has some number of battalions (sometimes one, sometimes two, sometimes four or more) which are the actual units used in military operations. So in this case the
1/43rd is the 1st Battalion of the 43rd Regiment of Foot.
A regiment with only one battalion would usually not include the 1/ before it, so this also means that the 43rd had two or more battalions (either at that point or at some nearby point in time). As it happens the
2/43rd existed from 1804 through to 1817, whereupon it was disbanded.
In 1881 lots of one-battalion regiments get merged together, which is not always for the good. This did remove most or all one-battalion regiments from the list though, so from then on any battalion would have a number (as there were always at least two battalions).
In this case the 43rd got merged with the 52nd (Oxfordshire) and produced the Oxfordshire Light Infantry, which became the Oxfordshire and Buckinghamshire only in 1908.
As it happens post-Cardwell the way of doing things was to expand existing regiments by adding more battalions rather than creating new regiments, at least in the infantry. In the case of the Royal Fusiliers, for example, there were at least 46 battalions raised during WW1.
As previously noted, regiments are administrative.
The 1/5th and 2/5th designations are a peculiarity that came about again in WW1.
I'll look at the complicated mess by following the West Surrey, the 2nd Regiment of Foot.
Essentially pre-WW1 the West Surrey (Queen's Royal Regiment, if you're fancy) had five battalions - 1st, 2nd (regular), 3rd (special reserve, which was the old "militia"), 4th and 5th (territorial force, which was the old "volunteers")
Once the territorial battalions had gone overseas the territorial associations - which were sort of enthusiastic - started raising additional battalions. These used the only battalion numbers (4th and 5th) that the territorials were permitted to have, but to distinguish them they became e.g. the 2/4th and 2/5th, the 3/4th and 3/5th and the 4/4th.
Then you have the New Army being formed, which added extra battalions to the regiment as well formed from new recruits - in this case there were the 6th, 7th, 8th, 10th and 11th (all "service" battalions which served overseas) plus the 9th and 12th (both "reserve" battalions which served in the UK training recruits). There were also several "labour" battalions (13th, 14th, 15th, 17th and 18th) which served overseas or at home providing construction labour, and a "home service" battalion (the 16th) which served for home defence.
Then late in the war it was permitted to form extra Territorial Battalions with new numbers. These were the 19th and 20th.
Finally, there were the 51st, 52nd and 53rd. These were a type of battalion used for training conscripts who had just turned eighteen - conscripts were first sent into a Young Soldier battalion (the 53rd) and then into a Graduated battalion (the 51st and 52nd). Basically the process meant about 200 new trained 19 year old soldiers every three months.
This is all terribly complicated, but here's the rule of thumb:
In the British Army, a regiment is administrative. If you hear about a formation of infantry in the field it will be a battalion.