No, however a number of other factors that led to the formation of the United States would still be operative. There would still be a desire for a common body to handle matters at a less remote and difficult level than London, such as coordinating tariff policy, the construction and integration of large-scale public works such as railroads and canals, and forming a common immigration policy. All of these were important factors in the formation of later federations such as those of Australia and Canada (and Australia, of course, was just as unthreatened by European events as North America would be), quite aside from any defense considerations.
Additionally, Britain switched to a more "unite-and-conquer" mode in relation to its settler colonies later in the 19th century (it encouraged the federation of Australia, for instance), and in any case is likely to want to delegate the burden of handling such petty disputes between New York and New Jersey to a lower level so that it can handle larger and more pressing issues than the question of who is legally allowed to run a ferry route between the colonies or what not.
This may not lead to the formation of a single unified North American dominion (and even if it did some peripheral colonies would likely opt out, as Newfoundland and New Zealand did), but it is likely to result in the federalization of many of the colonies into fewer and larger units than was the case in 1775.
Its called Parliament and lives in Westminster unless otherwise decided by Parliament.
Arguably the AWI was all about whether Westminster was supreme or whether the colonial assemblies were co equal with Westminster. If the British 'win' either by force of arms or compromise the end result is Westminster is supreme.
The basic post AWI colonial charter for New South Wales etc was form an assembly and pass any laws you like except no Laws repugnant to the Law of England and Wales with the Home Office Secretary being the responsible cabinet secretary until 1801 when its the War office. America is different to Australia in several ways, its a lot closer, 3 weeks sail not 6 months and has an established court system and its vastly different from say Malaysia as all the people in the American colonies are British. Models that aggregate various protectorates and really tiny colonies do not apply to British Subjects.
If the American colonies come together then like Australia it will be because they choose to or not as with New Zealand given the rivalry and different circumstances of the various areas I think thats very unlikely, slightly less likely than Third Act of Union with New England taking Seats at Westminster. The New England colonies having been part of British Society since at least the civil war.
So border disputes between the colonies are a matter for the Court of Chancery, I recommend the Ordnance Survey dispatch surveyors after the example of Messers Mason and Dixon to settle the matter but in the end if the locals cannot reach an agreement its a matter for the courts.
As to the army if you look at the US army which reflects the needs of a country suspicious of standing armies - like Britain is - then early on is likely to be a single infantry regiment, several detached companies ( as the previous Ranging companies) and an artillery battery. With a war scare, and there will be one assuming something like the French Revolution and Napoleonic wars that will increase but by the ACW the US army is 10 IR 4 Arty Rgt,( say 36 Batteries but that includes sea coast) 2 cavalry 2 dragoon and 3 Mounted rifles. The British army as at 1855 was 108 Inf Rgt, 26 Cavalry regiments and 199 arty Bty, Plus the East India Company regiments, at 154,000 + men. So it seems in keeping with the scale of things up to then.
By then the small number of IR will have numerical designations, plus a semi official nickname, the 60th will be the American Rifles and there may be more than one regiment so described ( don't forget a regular regiment is available for service anywhere, usually Martinique) , 104th the Royal Newfoundland Regiment. and the others I would guess would include some called Rangers ( its an old designation) based on 1812 Canadian units a Frontier Light Infantry and Western Rangers seem likely but as nicknames.
Come Childers reforms those will get regional designations or have the nickname made official. The Loyals (a British army regiment) were originally the Loyal Lincolnshire Volunteers, a regiment of volunteers raised at the time of the French Revolution and later incorporated as the 81st foot. then merged with the 43rd.
The Royal designation has to be earned so some of the older regiments may get it or Kings/ Queens Own but its not automatic or necessarily welcome.
What a war scare will do is raise a number of regiments for Home defense initially which if the scare is closely after the revolutionary period may reflect whatever units were raised then. So based on that and the Rule of Cool,
The Ethiopian Regiment aka the Black Banditti, Georgia Regiment, Kings American Regiment, Kings American Dragoons, Loyal Americans, McAlpins regiment ( fusiliers obviously ) Prince of Wales American Regiment, New Jersey Volunteers, New York Volunteers ( and so forth across the colonies/provinces.) Queens American Rangers, Virginia Light Horse, Canadian Voltigeurs, Provincial Marine ( Great Lakes not quite navy)
And if the colonies are designated Provinces rather than States a lot of Provincial formations - Provincial Guides and Pioneers. Things get weird when the volunteer rifle corps take off. Port of New York Submarine Miners for example.
