I will be having the following arrival and departure times in and out of LHR.

Arrival at LHR: Sunday, 7am
Departure from LHR: Monday, 8pm

So, check-in could be as early as 9am on Sunday and check-out on Monday as late as 5pm.

Do you have any ideas on how to maximize hotel room time without having to buy for two or three hotel nights?

I've heard of hotel loyalty programs that, given a certain high status, allow a 24-hour stay at the cost of one hotel night, but I forgot which ones do that? Which ones are those?

I have mid-tier status with several hotels. Which ones are most known to allow very early check-in and very late check-outs?

I'd like to stay in central London (maybe near Piccadilly Circus) and my hotel budget is probably GBP 150 - 300 for the entire stay.