Hi Neil,
Generally speaking, most British and American regiments had eight center companies each. With very few exceptions (like the 71st Highland Rgt.), regiments had only one battalion. If you choose to depict 320-man battalions on your tabletop, you'll end up with 16-figure battalions for both sides, using the 1:20 figure scale.
Paper strength for the regular units of both sides called for more than 320 men. But that seems to be a good ball-park figure that represents a lot of units at a strong field strength. You'd certainly be justified in making battalions smaller than 16 figures, especially for the Americans.
Both sides had elite companies, which were just about always removed from their parent battalions and put together in composite battalions. The British had light and grenadiers companies; the Americans had only light companies (although a handful of American grenadier companies did exist, I personally wouldn't bother with them.) The size of the elite company formations varied, depending on the amount of companies and their strengths. Burgoyne's elite battalions, for instance, had ten companies each.
There were some exceptional British formations, such as the battalions of the Highland regiments and the composite Guards battalions, which could be depicted with 24-man battalions.
The German troops, such as the soldiers of Hesse-Kassel, had regiments/battalions of five companies: four of musketeers or fusiliers, and one of grenadiers. The grenadiers were detached from their parent regiments and formed into four-company grenadier battalions. The German regiments could be depicted with 20 or 24 line infantry figures (that is, four companies with either five or six men each.) The jaegers (at least those of Hesse-Kassel) often fought in single companies, again using five to six miniature figures. Incidentally, there was no difference between Hessian musketeer and fusilier regiments: both fought as line infantry.
American militia formations seemed to vary in size, and I'm not sure if one could choose an average size. You could simply paint up a bunch of American militiamen, with a handful of command figures, and thus make miniature militia battalions of whatever size you wanted or needed.
What I've written is pretty general, and no doubt subject to correction! Hopefully this will give you a starting point. I'd like to recommend that you check out the Greg Novak books, which are published by Old Glory: "The American War of Independence: A Guide to the Armies of the American War of Independence." Book One covers the northern campaigns, Book Two the southern. These books are the most comprehensive, one-stop source of information on American Revolutionary armies and orders of battle that I can think of.