the french army of early world war 1 OTL wore blue jackets and red pants so for an alt-UK army, you could have ww1 type uniform but with the jacket being red and dark grey pants for light infantry and other non-khaki colours depending on the branch,
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M. Pasquin, this is a worthy concept in its own right - rather horrible to contemplate a Government & High Command foolhardy enough to send redcoats against machine guns, even worse to recall that the French Army actually did something so wilfully stupid in Real Life out of wounded National Pride.
It would definitely make sense to homage the 'tin hat' of World War I & II by (at the very least) showing the local redcoats use round hats of the sort frequently worn during the American Revolution to mimic the silhouette of Tommy Atkins in the trenches - and showing them use puttees would probably be quite workable too, though gaitered 'American trousers' might be more in keeping with the 18th century notions of style on which this setting is built.
For the record, my notion of this setting is that it has arms technology roughly equivalent to the early-to-mid 19th century; still rich with smoothbore 'Napoleons' and muskets, but with percussion caps rather than firelocks (and with rifle-muskets & rifled cannon increasingly coming into use). Steam-driven technology either absent or fabulously obscure (so canals, rather than railroads and steam-driven ships are cutting edge to the point of being experimental at best), but hot air balloons see some use and binoculars are not uncommon.
In terms of British Army uniforms, an adaptation (or illustration) of the original would probably have to deal with two specific periods - one more briefly than the other - to wit, the Year 1900 (In which His Majesty assumes the throne and is then obliged to kick the Russian invaders off British soil, then right back to the other side of the North Sea, as you do), at which point the army and (to a lesser extent) the navy are likely to show the consequences of false economies inflicted in the interest of coping with the consequences of debt by the mint-load, and the Years of the Great War (if memory serves from 1919-1921), by which point The King has had the better part of a quarter century to bring his forces into a condition of some splendour.
My assumption is that the infantry would likely wear red coats with blue trousers (a nod to the dress uniforms of the pre-WWI period), with black cocked or round hats, brown or black leather equipment: boots likely brown or black leather. Facings vary by regiment, but regimental lace may be limited to plain white (with only the shape of the lace changing between regiments). Hair still worn long in a que, but powdered only on parade. Not sure if grenadiers should wear bearskins or cloth mitre caps - the latter offer more opportunities to display regimental badges, if nothing else.
Perhaps fusiliers wear bearskins, grenadiers mitre caps?
Artillery still bluecoats, cavalry likely follow the heavy cavalry = redcoat, light cavalry = bluejacket paradigm (Not sure if there ought to be lancer regiments in this British Army, but there has explicitly been at least one long war against Russia, fought during the 19th century, so its possible the Poles fought alongside the British Army long enough to make an impression).
At this point one realises that I should almost certainly look up British army full dress uniforms from just prior to the Great War and consider how one can filter them through a late 19th-century aesthetic (probably from the 1790s, just before the French Revolution & associated war-making); I also ought to look through the book once again and work out just how many armies are actually involved in the diverse conflicts (So far as I can remember the British, Russians, French, Austro-Prussians*, Kingdom of Italy as formed by the Two Sicilies, the Kingdom of Venice**, the Electorates of Hanover & Bavaria, the Swiss Confederation and the Spanish are all mentioned - as are American Provincials from His Britannic Majesty's Province of Louisiana).
Things to do, things to do ...
*No, really - a Hohenzollern appears to have emulated Henri IV sometime in the 19th century ("The Reich is worth the rosary" ya?).
**Clearly somebody pulled a Medici on
la Serenissima.