Overnight layover at Heathrow

There are London tours at night but I can't recommend one over the other.

Not sure if they go near Paddington as the station isn't exactly in the center of everything. You may have to take the tube to a closer area of the city to catch the tour.

Just realize that if your plane arrives at 5:30, that by the time you get through passport control and check into the Sofitel, it could be close to 7:00 when you start your journey from Heathrow, Just be aware of the latest trains back.

This is definitely doable. And in June it won't start to get dark until after 9 PM.

London Walks does "pub crawl" tours at night
http://www.walks.com/

I'm not sure what's on offer the night you'll be there, but in years past they had a "Ghosts of The City" tour and a "Jack the Ripper" tour, both with stops at a few pubs along the way.

You won't find an organized tour from Paddington. Trip Advisor shows this tour, leaving near the Green Park tube station at 7:30 and every half-hour thereafter, and taking about 90 minutes. The link is to Viator, which is a booking agency not the company actually offering the tour. Further googling might help you find that tour company itself so you could book directly with them instead of giving Viator a cut of the price. It might also reveal other such tours. Looks like all you do is sit on the bus, listen to a narration (live? recorded?) and look at the lit-up city, probably not time to get off anywhere.

If you take the Heathrow Express to Paddington you'll have to taxi or tube to Green Park. The tube would involve a transfer at Baker Street, so allow plenty of time. It might be more efficient to take the Piccadilly Line straight from Terminal 5 to Green Park. Either method will take close to an hour. The Transport for London site can help you plan. Keep in mind that the last Heathrow Express leaves Paddington at 11:23, and the Piccadilly Line will also stop or run very infrequently late at night.

If you end up at the Sofitel and don't go into London, you can have a wonderful dinner at Sofitel's restaurant Le Belle Époque. We always enjoy dining there.

You might be able to make the 8:00pm bus tour but it will be tight.

http://www.hoponhopoffplus.com/london-by-night-open-top-bus-tour

If you can't make it to the bus tour, you could rush over to the London Eye (open until 8:30pm). I think the Eye is best at night but that won't be possible in June...

I nice walk on the south embankment and a late evening pub meal would also be an idea.

Keep in mind you'll need to get some British currency which can be accomplished at Paddington by using a debit card at a bank ATM.

Instead of having to negotiate the tube check the tfl.gov.uk website and see if the 28 bus from Paddington to Trafalgar Square/Whitehall has a night route.

If yes take it to Trafalgar Square, get off, smile and start your stroll down Whitehall from there to Parliament Square.

Pretty simple route to follow, merely head for Big Ben.

Enjoy.

Tom, I actually plan my flights to intentionally stay overnight at Heathrow, at the Sofitel, if possible. BA does not require me to stay overnight. It makes airline travel more bearable, and provides some lovely respite to the travel. You can usually time flights to avoid this if you'd rather have a really long day or an overnight flight. That's not for me if I can help it.