I think Illa's assessment is the truest and most accurate assessment so far. That doesn't mean I'm going to stop writing. I'll still provide my opinion to satisfy my ego.
Meanwhile, I greatly disagree with Devilfish on a couple points. But I will probably get to that later.
British - RAFProbably the most popular brit doc because of its fluff. It's a slow starter with low early game power, is okay mid game, but late game is absolutely punishing. One tip about playing RAF, one of the best units available for Brits in general is the Recce and its insane sight plus durability. Similar to Airborne doc and among british docs in general, RAF is probably the most versatile making it an extremely power doctrine to play in 1v1's or "team carry" scenarios. That doesn't meant there are tradeoffs though.
Army Composition - Balanced, like AB for USA it's a well balanced doctrine as long as you can make the right doctrine choices to field the units you need when you need them. There is a lot of decision making, so I recommend to not commit to a particular branch unless you are covered by your teammates or have found a way to cover yourself.
Micro Intensity - High, at least until you can get into the late game where your soldiers are extremely durable and you have many "click to kill" ability options available through doctrine unlocks. At that point, you're more or less and unstoppable force walking through the park.
Weakness - Early game you lack any extra advantage than any other brit doctrine can have and need to rely a lot on teammates. Your build order might be a bit rigid in order to hedge against potential enemy openings. Brits really don't have very strong opening moves in general except to get AT to hedge against car rushes (if you don't and they do get cars, you're screwed. if you do and they don't, you might have to fight harder with weaker units but at least you can survive the early game)
British - Royal EngineersProbably second most popular doctrine for Brits in PvP because of how forgiving the doctrine can be (as long as the doctrine is played right). The durability of the units, resource bonuses, and unit costs really don't make decisions that hard. Often times you'll find the only resource you'll be short in consistently is manpower. RE are weak early and extremely strong mid game, start to taper off and get harder to play as the game drags on. Biggest tip: DONT FORGET THE DOC IS MORE THAN JUST CHURCHILLS! Using base brit units are really how you can cover the doc's weaknesses.
Army Composition - pretty balanced, the inf are naturally strong as Brits, I'd recommend not treating the engineer glider like RAF gliders but there are some players who do use them like shock units. The churchhills are definitely strongest mid game, but become less powerful as axis can field stronger AT and tanks. Churchhills make great line pushers but lack the damage you'd find in most other tanks. mainly because their best range is mid to close but the churchhills have no gap closing mobility except to pray nothing penetrates as they crawl up towards the target. Lastly there is some light arty available for special situations, not as plentiful as other docs but it's enough to be considered "balanced" in my books.
Micro Intensity - Medium, RE is more about strategy than micro, imo. The units are too slow to truly micro except maybe the shock royal engineers which is even more micro intensive than average because they lack hold ground (soon to change) and are extremely squishy. The real micro comes in when you use smaller cars like staghounds and daimlers which really help for things like hunting mobile arty and mortar cars (i really don't know why more players don't do this...). But, that isn't far different than other factions.
Weakness - panzer shreks and overconfidence. Pre 4.8.3 patch, people complained about how useless RE was. then churchills were buffed and people complained how strong they were because no one was use to having any other AT tactic other than plopping a simple pak firing the front of a tank. since then people have adapted and found other ways of killing the churchhills, so don't get over confident with you durability. Use it well and strategically, pair it with your teammates, and you will be an amazing supporting player where people will start calling you out for being OP. The most glaring weakness is just lack in mobile differentiation from other factions/doctrines.
British - Royal ArtilleryProbably most popular among compstompers I think? I always see people who play compstomp and jump into PvP either play RAF or Arty. Personally, I think the supercharge mortar/arty rounds is one of the strongest doc techs in all factions/doctrines (first place being recon abilities). This doc is all about playing strategic distance. Keep an enemy out of their striking range while throwing your long range at them. Biggest personal tip is to not build a static defense. You'll just get recon'd, counter-bombed, and have no ability to strike out against their long range units (more on dynamic defense here
viewtopic.php?f=30&t=1153). This is tough because British lack mobility as is, but it is even more of a reason to make sure you keep shuffling your defenses so that your opponent can't lock on to you. Choose mobile mg's and AT over the emplaced, decide to go on the offensive not to kill but purely to keep the enemy at bay, moves like that will protect your mortars and arty to do the real damage dealing. Use the spotter. Long ago, the dingo couldn't camoflauge and the spotter was the only invis recon Brits had. Spotters are cheap alternatives to the dingo.
Army Composition - Low, relative to all other doctrines and factions there really is not any unit variety gained from artillery doc except more artillery. Arty doc doesn't even have the best arty units, it just has a lot of it. Quantity over Quality.
Micro Intensity - Low to High, so this is really team dependent. Without teammates, you need to cover yourself with non-doctrine units, carve out a spot for your artillery, then finally secure the area for artillery before you can even think about building artillery/mortars. With non-specialized units from base British roster, you will probably have to push your micro skills to the limit to reach an arty ready foundation. If you can have teammates to keep the enemy at bay so you can just deal the damage from afar, you're fine. Annecdote: On that god awful map halfaya pass, I spent 70% of a game alt tabbed while my arty was on cooldown, only to come back to play to fire off more rounds. Came out with the highest score, doubling the second place highest score, least amount of losses, and wasn't even actively playing the game. This is an extreme scenario given circumstances... but just an example of how low micro intensity could be.
Weakness - Highly strategic and micro dependent to really use the doc effectively. You get no special frontline units and have to play with only a few non-doc roster units because of British high manpower costs. Repeated attacks where you at least win cost-cost against an arty doc player will eventually break them. As an arty player, make sure you always win cost-cost or you'll keep falling behind.