London and Munich hotel recs

Do you have any status with any chain? Or an amex/bilt/chase card that you can stack benefits on?

Being close to the airports means you're not close to any tourist sites (except Windsor Castle in London).

Budget range? I liked the London Edition, but it's 420 pounds a night.

The tube in London is very effective. Just find one with a tube entrance nearby and you will be fine. Worry more about not lugging all of your luggage onto the tube during rush hour if possible

For both airports just do the 1 hour public transit each way and stay in the city. Not carrying luggage is a good idea - look into stowing your large bags at the airport.

Platzl Hotel in Munich is very nice and in a great location
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For Heathrow - you can also take the Elizabeth line into town if you don't want to spend the extra $$$ on the Heathrow Express. Elizabeth is the newest line - nice cars and air con - just five minutes or so slower than the HE, but will save you 10 pounds per person at least. Elizabeth line trains also run more often than HE.

Wife and I have stayed at The Royal Horeseguards multiple times. Right on River Thames and two Underground stations nearby. Very close to Big Ben, Tower of London area.

If you want to be "close" to Heathrow and yet enjoy London and not zone 5 or whatever quasi-suburban LHR is.... Stay at the Paddington Hilton. It's a 15 minute ride on the Heathrow Express train (not the tube, taht's an hour or so) and you're in zone 1. You can then walk or tube in 10-20 minutes to the restaurants, Pall Mall, City, etc...

That hotel can get pricey on busy weekends, there are a few other clean lower cost places like the Dolphin five minutes away.

Finally if you land after an overnight, Raffles restaurant has a passable "full english" breakfast that will reinvigorate you.