isnt a TM30 different to the 90 day notification (TM47? seems to me that he needs to notify of being in country for 90days - that is independent of the TM30 housemaster form.
as long as you remain in transit and your luggage will be checked through to London on same ticket, is possible. If you need to do a self-transfer (ie checkin luggage on a new flight) then need new TP, insurance, PCR test/hotel etc.
Heading from where to where? Domestic connection? General speaking you will need to do either test n go or sandbox. Test n go can do just one night then travel elsewhere.