Commenting on the opportunity to do a LHR shuffle, at LHR T4 is closed which is where the Qatar lounge is located, and is expected to remain closed for some time, and if rumours are true will probably be the last terminal to reopen at Heathrow. Qatar have just relocated from T2 to T5 to co-locate with BA and an increasing number of other OW airlines and if air traffic does not recover they will likely remain there for quite some while.
The Qatar lounge at Heathrow is probably off limits for quite some while...