24 hour layover in London - what to do?

Some info from TA:
https://www.tripadvisor.co.uk/Travel-g186338-c170832/London:United-Kingdom:Between.Flights.At.Heathrow.html

Assuming you arrive at Heathrow.

Are you staying overnight? If yes the best option to see the iconic sites is to tube to Westminster. Walk out to Parliament Square ( Houses of Parliament with Elizabeth’s Tower and the Big Ben Bell ( totally in scaffolding till 2020) and adjacent Westminster Abbey.

Walk mid span on Westminster bridge for photos of Parliament on one side of the Thames and the London Eye on the other. Walk back to Parliament Square and up Birdcage Walk through St James park to Buckingham Palace, Walk through Green Park to Piccadilly Circus down Piccadlly to Trafalgar Square.

Grab a nosh at the nearby Sherlock Holmes Pub then walk along the Thames path back to Parliament Square and tube back to your accommodation.

If not staying the night you will need to figure out what to do with your luggage.

Others will suggest taking the nearly hour tube ride to Victoria station which involves getting off at Hammersmith then boarding another train to Victoria Station. Not me. Useage of you Time is a focal point.

I suggest the following.

Pre book a duo fare on the Heathrow Express. It’s a 15 minute ride to Paddington Station. Exit and walk outside, head past the Hilton hotel to the HSBC Bank ATM and withdraw a few pounds. Walk across the road past the pride of Paddington pub and walk a few blocks over to Lancaster Gate Station ( you can glimpse Hyde Parks Italian Gardens across the road). Use an automated machine to buy your Oyster cards. Hop on a central line train to Oxford Circus, Mind The Gap exit that train and then follow signage take a bakerloo train to Trafalgar Square/Charing cross.

Get out and explore Trafalgar Square walk down Whitehall toward Elizabeth’s Tower to Parliament Square. Follow the route up to the Palace, Over through green Park to Piccadilly Circus.
Bakerloo line to Oxford Circus. Get on a central line train to Lancaster Gate. Walk back to Paddington Station and hop on the Heathrow Express back LHR.

Use google maps satellite view to review what I just suggested.

Don’t know when you are traveling but look online about the Heathrow Connect which changed its name in May.. Might be able to ride it by using your Oyster Card which you’d need to buy at Heathrow.
Think I read this happens in September but best you look up the information yourselves.

Have fun.

One can see a number of sights in the time you’ll be there, but I consider it the whirlwind tour since you won’t be able to spend a lot of time at any one place. All the major sights are accessible via the tube, so make a list of those you’d like to see, then put them in an order where you can see them yet not have to double back. Where you start will be dependent on where you’re staying. I took a coworker on a whirlwind tour a couple years ago and we stopped at Buckingham palace, Westminster Abbey Tower of London and Tower Bridge, Blackfriars pub, Parliament, MI-6 building, and Monument. You may have time to go into one place, so choose wisely. In the evening we hung out at the Blackbird pub by the Earl’s Court tube stop sine it was close to our hotel.

What day are you going to be there? Many of the museums have a day that they're open late, so that might allow you to see two sights. There's also theatre at night, so that could be an option as well.

Hello, all. Thank you for the suggestions. We will be there the third week March, mid week. We will be staying the night - any recommendations near the big sights?

If you are considering my advice regarding the Heathrow Express or Connect ( or whatever the new name is) then stay near Paddington Station. If you have points there's a Hilton or by using the search engine you can see other hotel recommendations near Paddington that have been posted by others. Hotel Indigo seems to be popular.

With your Oyster card and access to the Central Line at Lancaster gate Station you can be anywhere in a matter of minutes.