If you are against 3 OKW players, I suggest that one of the Brit players with good micro get double snipers since they won't have any counter to them. Lay mines down at the entrances to key buildings so they lose their falls or JLI that try to get behind your snipers.
Also, one of the British players should also fast tech cromwells. No bolster infantry, no Bofors tech, just straight to cromwells. 3 OKW players means no Non-doctrinal snares which means your cromwells can crush their infantry with impunity. After your second Cromwell go hammer for the warspeed which makes crushing easier and getting out easier. You can tech boslter squads and racks after that. Most allied players don't like getting a lot of tanks vs OKW BC they have shreks but Cromwells are fast enough to crush and get out if need be. Your Cromwell should be able to come out just after the Luchs. Most team game OKW players don't get puma because they think they can hold out for heavier armor.
This strategy has gotten me to top 3 for brit 3v3 random and 4v4 random. It works if you can keep your croms and snipers alive. Basically you'll be constantly bleeding then of MP while hardly bleeding yourself. Snipers + croms make Obers and all other OKW elite infantry fairly useless BC they can't get close to your snipers and can't dmg croms. Also, cromwells are excellent for hunting down stukas and ISGs which are what Brits struggle with most. Once enemy gets KT, get a FF with tulips, crit shot the KT with sniper and surround him with croms. If KT has infantry support, use a cromwell to crush them.
Also, vanguard regiment + commando demo charges is very useful. This will replace one of your snipers and maybe ISs with commandos and airlanding officer. End game croc seals the deal vs volks blobs.
In summary, Brits must use snipers and croms to counter OKW