I have just transited LHR T5, on BA Schengen to Schengen. At checkin in WAW I was not asked anything about a UK ETA. Apart from a boarding pass scan for terminal access in T5 there was no other check. A week earlier I did the same thing with QR through T4 on my way from CPT to WAW.
Note: I do have an ETA on my passport and it is possible that everything from CPT and WAW checkin to T4/T5 transiting was clever enough to know that, but I doubt it. So my takeaway is it is not needed to transit airside at LHR, regardless of terminal.
I am going to ‘transit’ landside tomorrow using the egate/First wing (hopefully quicker) so will see what I am asked, if anything.