Layover (7 h) with self-transfer from Heathrow to Gatwick

Depending on when you arrive you may want to consider this option.

https://www.londontoolkit.com/tours/heathrow_windsor_tour.htm

Personally I would not tempt fate by taking a detour between airports.

Don't expect to stop anywhere to get some sightseeing in. You might be able to book a car to drive you through London between the two airports rather than go direct. You might see a few things through the window.

Traffic in London is terrible and this kind of detour will take a lot of time. Don't plan to make stops.

What day of the week is this? What time does your flight arrive?

If you arrive in the morning and on a weekday, you'll be fighting rush hour traffic to get into the city.

Yeah, I would advise seeing this less as a “layover” and more as a “thank goodness we’ve got plenty of time to make the connection.”

The normal road route would be via the M25, where all you’d see is traffic jams.

One of the reputable car firms such as Blackberry or Just Airports or Addison Lee might be able to quote for a journey between the two via central London with views of some landmarks.

You need 1h30m for the transfer, and if it's long haul that should be aiming for 3 hours before the flight. So that gives you 2h30m. I really wouldn't bother, it's way too rushed and a you'll spend more time finding somewhere to leave bags.

And if your trip involves Gatwick, I really wouldn't take a taxi out of London. Either use the coach directly between the airports or Piccadilly line to Baron's Court, District line to Victoria, train to Gatwick

Whilst you have enough time for the transfer you can't add any meaningful sightseeing in central London.
What I would do is take the train. That way you will at least see some fleeting images of the city. The new Elizabeth Line makes this much more practical than it once was and it's now one simple change. Elizabeth Line to Farringdon and Thameslink train to Gatwick. Whilst the Elizabeth Line goes underground for the last but from Paddington the Thameslink train is mostly above ground and you will get a glimpse of the Thames around Blackfriars. It should take about 90 minutes and won't be subject to the vagaries of traffic on the M25.

We did this transfer by National Express coach direct from Heathrow to Gatwick.

You are kidding yourself if you think this is an opportunity to see London. Come back when you really have time.

Unless you have a +24 hour layover you need to save seeing London for another time. All you need is for your plane that is arriving into Heathrow to be late to make the transfer to Gatwick a lot more interesting. Also, remember you have passport control etc to deal with at LHR.

4 folks, 4 people's luggage to wait for on the belt. An hour to get to whatever transport. One and a half to two to join the two airports. 2 hour or 3 hour check-in depending on airline and destination (you haven't said).

I make that (pessimist that I am) 6 hours. One hour isn't enough to do anything.

For glimpses the Elizabeth Line and Thameslink I agree. West London before diving underground, Londoners going about their day and views of the stations. Then the fairly seamless connection to the Thameslink, views of St Pauls (look left (facing forward) and back over your left shoulder while the train stops at Blackfriars for a glimpse of the River and St Pauls) and then South London railway industrial for a while until you get into the rural area, then off at Gatwick (the train only stops a short while - don't dawdle) and upstairs into the airport. Figure out your terminal before you arrive. There is free transport every 90 seconds from South Terminal to North Terminal. Put your feet up and get a coffee.