Yes, both BA12 and BA16 from SIN-LHR arrive at Terminal 3 until sometime next year -- I also seem to recall sometime in April being the changeover date.
If arriving at Terminal 3, the OP's flight arrives well before the AA arrivals lounge opens -- both the BA and AA website state that it opens at 6:00AM. So it would appear the only option would be to clear immigration/baggage claim/customs and hop on the Heathrow Express (free when transferring terminals) over to Terminal 5 and use BA's arrivals lounge which opens at 5:00AM.
BA's arrivals lounge at T5 is pretty good from my experience -- I'd recommend it even with the hassle of having to transfer terminals. I've never experienced queues for either getting in or using showers (they have 60 or so showers available!), or with overcrowding. They'll even press your garments for you while you shower! The lounge shouldn't be too hard to find since it appears on the signage in the landside area of Terminal 5; when you'll in the main hall, just look for an orange-colored lift and take it to the first floor.