12 hour Heathrow layover

Which budget airlines fly out of Heathrow?

I didn't think any did and I can't find any on their website.

plane lands 9:50pm, out about an hour later - still time to get the last tube in.

take off 9:50 am, need to be back 2 or maybe three hours before, takes about an hour from central London. If it is a Sunday I don't think this works, otherwise ok.

look at tfl.gov.uk. Oh, and make sure your pack fits in with cabin luggage dimensions.

and I don't think there are any budget airlines at Heathrow.

There are NO budget airlines and NO budget buses from Heathrow. The tube (metro) is as budget as it gets.
But London after hours is not fun. You will not know where you can safely go.
And there are a LOT of drunks, vagabonds and crack-heads wandering the streets after midnight.
It is not fun ending up in a hospital emergency department at best, a lead-lined coffin at worst.
Read [www.sleepinginairports.com] Stay at the airport, and get to where you're going.

Is there anywhere to go besides Central London? A little pub?

Most little pubs will shut at 11pm.