The AA arrivals lounge is competitive with BA but not quite as nice; however good enough and the most important feature, the shower suites, are just as good as BA offers (clothes pressing is also available though I've not used that service). The breakfast is continental but has some of the best Danish in the UK, and the coffee is excellent. Internet terminals are much faster than BA offers. Most importantly there are few pax entitled to use the AA facility, so it is generally less crowded despite being smaller. There is no MB spa equivalent, however the arrivals MB spas in T4 and T1 both offer an abbreviated massage (you only get the full massage when departing -- an interesting strategy!), also quite hard to get a spa booking during peak times.

Note that the BA departure lounges at LHR all have shower facilities available (even the T1 UK domestic lounge does, albeit not as nice as the T1 international lounge).