What to do with my 10-hour layover at Heathrow?

"You are not really going to find a true 'English pub' in tourist central"
That's not true. Look these up beforehand. They are all within 20mins walk of Westminster Abbey:
St Steven's Tavern - A mix of tourists and the Westminster crowd
Gordon's Wine Bar - Not a traditional pub, but quirky and good for food (cold buffet of meats and cheese)
Lamb and Flag - one of the oldest pubs in London. Classic British pub (20min walk)
French House - Soho. No food but friendly. (20min Walk)
Fleet Street has some great boozers. Really old ones. Ye Olde Cheshire Cheese for example. A nice walk would be from Westminster, cross bridge, along South Bank to Tate Modern, cross back to north side of river and head back West to Fleet Street.
https://maps.google.com/maps?saddr=Bridge+St%2FA302&daddr=51.508478,-0.101195+to:Fleet+St&hl=en&ll=51.507193,-0.10922&spn=0.01373,0.054588&sll=51.513854,-0.108308&sspn=0.001716,0.006824&geocode=FYjXEQMdnxj-_w%3BFf70EQMdtXT-_ykpnSGVrgR2SDHA434OjNTcvA%3BFWYLEgMdZF3-_w&oq=lamb+and</i>flag&dirflg=w&mra=dme&mrsp=2&sz=18&via=1&t=m&z=15</a>

Note - Fleet street is further than 20mins walk!