My info is slightly dated but as others have said:
T2/3 are the same stop, (aka "Central Area" on some signs).
There's certainly the Underground option(contactless card).
Another option might be Heathrow Express - it goes through T2/3 on its way to London and you certainly used to be able transfer on it for free with either a contactless card or by using one of the free tickets available from either the staff or from one of the dispensers on the way onto the platform.
FYI and in case it gets mentioned down thread the TfL red buses that were also really handy for getting around the airport are not free anymore.
Back at T5 the arrivals level itself can be cold and not very welcoming, though there's an M&S food store if you want snacks..If you do decide to loiter in T5 then it might be worth going up to the departures level - top floor - it's a bit warmer up top and there are, or at least were, pre-Covid, a couple of restaurants "landside" (sadly Carluccios of course is now gone) plus a coffee shop at the northern end of the departures level. there used to be a reasonable amount of seating..but given Covid?
As I recall the bus station T2/T3 Central area bus station is a bit cramped and not very welcoming but maybe it has changed since I last used it.
Good Luck.