I am assuming that Mitchel and Tyler was the name of the Richmond retailer, not the British watch maker. Thus I fear we would need to have access to the watch to determine much more about it. Your chronometer watch itself likely did not carry the retailer's name, although it could have. Much more likely, though, is that it carried only the name of the English manufacturer. The case, unless the movement was cased in the US (some English watches were), should carry an assay hallmark (most likely, London), a date mark, and either a sterling silver mark (a rampant lion, indicating 92.5% purity), or a gold mark indicating the gold purity. These marks will be on the inside of the rear lid. There will also be a two-initial maker's mark on a lid interior or on the pendant, as the case and the movement would have been made in different shops. Most English case maker's marks can be identified to specific persons, if one knows the production date. English cases will be "swing-out," meaning that one must trip a catch on the edge of the dial (usually at 6 o'clock) with one's fingernail and rotate the movement up out of its case on a hinge. To see the movement, and the chronometer escapement, one would then need to release the latch on the integral dust cover and remove it. Unless the watch is a private label for the US retailer, you would then be able to see who actually made it. Production information is available for some English makers.
As you may know, in the 1859 period, the term " pocket chronometer" referred not just to a well made, relatively accurate watch, but to one with a "chronometer escapement." For those who may be less familiar with chronometers than yourself, allow me to explain that there were 2 basic types of chronometer escapement, and numerous subvarieties. "Spring detent escapement" pocket chronometers were favored by the high end English makers (such as Charles Frodsham, to name one, or Dent, to name another). The John Arnold marine chronometer (not a watch, but a clock) and the copy of it made by Larkum Kendall, which ultimately won for Arnold Parliament's coveted Longitude Prize, also had a spring detent escapement. (This is in fact why detent escapements were called "chronometer escapements." Marine chronometers, on gimbaled, shock absorbing mounts, and which were used on ships for determining longitude, had spring detent escapements as well.) The Swiss favored "pivoted detent escapement" pocket chronometers. No American makers were yet making pocket chronometers in 1859. (George P. Reed, who ultimately made perhaps a hundred or more pocket chronometers of his own patented designs, was to be America's most prolific maker, but he made his first chronometers around the time the ACW was ending.) The majority of pocket chronometers entering the US, which weren't many ($325 was an enormous sum for a watch in 1859!), were English, and especially so in the South, given that antebellum southerners idolized and obsessed over things English. Watch movements with chronometer escapements will have temperature compensated two-armed bimetallic balances (so called "chronometer balances") with gold timing screws. The mean time screws nearest the balance arms will have threaded nuts on them. The hairspring, which attaches the balance wheel to the balance cock, most likely will be helical (i.e., cylindrical), as helical hairsprings were the most "isochronous." (Most watches of the period had spiral (aka "volute"), disc-shaped hairsprings, which were cheaper to make, but gave more of a variation in rate over the running period.)
Here is an example of an English spring detent pocket chronometer by Thomas Rushton in a classic open face English Sterling silver swing-out case with the London hallmark and the date mark for 1839. This example is sufficiently early that it has trapezoidal weights on the balance wheel instead of timing screws.
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Probably something that looked a lot like this (center second hand and lots of gradations), in a similar 18K gold case (English, of course) :
Bringing the