Is a 13-hour layover in the middle of the night enough time to go see Buckingham Palace?

Yes, it's doable. Although going through central London isn't the fastest route from Heathrow to Gatwick, you have lots of time so it's not an unreasonable route to take.

You could obviously do it all by taxi, but it's also possible (and much cheaper) on public transport. You can get the underground from Heathrow to Green Park, which will take about 50 min, and then you're only a few minutes walk across the park to the palace. You can then walk to Victoria station, and get a train to Gatwick. The last train to Gatwick is at 00:08 (though trains from other stations and coaches run through the night, and the ones from Victoria start pretty early in the morning).

Very easily doable. But if you're going all that way in on the tube and aren't too tired, walk around - central London is quite small and easily walkable. You could see Tower Bridge, Tower of London, Big Ben, Trafalgar Square and Picadilly Circus - all quite photogenic and iconic, even at night!

Yes, with Caveats. Budget for contingency - 2 hours on landing, maybe 3 for T5. I'm not kidding about T5 at this time of year. A morning arrival might have been better, it will be dark.

So everything else is down to visibility, you will miss peak hours. Suggest you get a driver both ways, or tube and cab. Buses do not pass the Palace. You are taking your chances with visibility as well. Plan to be back at the Airport at 06:00, again, for contingency.

Risks: Snow, Tube Outage, Rain, Freezing Rain, Sleet, Traffic, Holiday events (Winter Wonderland is close by).

Per other responses, the short answer is "yes you do" but you're only going to see the outside of it.

A brief stroll back up the mall will land you at Nelson's Column, and you're near enough some night-life such as Leicester Square, or the theatres on the embankment, as well as within wandering distance (or a short tube ride) of the illuminations in Carnaby Street (Bohemian Rhapsody this year!) and there's doubtless a few others to see too.

I don't know the opening times but over the river you've got the London Eye which could provide a decent photo op too. Just a shame Big Ben is scaffolded up for repairs for the forseeable.

It takes 45 minutes to get out of the airport and another hour to get from Heathrow to BP. You'll arrive at BP at 10.15pm. Not much to see at there at that time, so I think you'd do better to head over to Waterloo Bridge and then wander up in to Covent Garden, Leicester Square, Picaddilly Circus, and Soho. The cheapest way to get to Gatwick is by coach from Victoria Coach Station. It's a miserable trip, early in the morning, but it sure is cheap.