Thanks for all the responses which proved extremely helpful. I'll like to summarize my experience:

Leaving out the expensive HEX option (with a few passengers you might as well take a cab), the best train+tube route from the East Croydon station to LHR was a regular (non-tube!) train to Victoria station, then District line, then change to Piccadilly line.

The Victoria station had a set of unavoidable stairs, surprising for such a busy commuter station. Change between District and Piccadilly lines was smooth, just across the platform, at both Baron Ct and Hammersmith stations (maybe there are others as well, but these two for sure).

However, lesson for the future: While we enjoyed our research and our journey, all considered I feel that the bus X26 would have been our best option.

Somehow, to our hosts as well as myself "bus" had sounded automatically slow and "train" automatically fast, but the fact that we had to take three trains, go from one platform to another and then wait for the next one, negated the train's speed advantage. X26 would have taken us from E Croydon station to LHR directly and any time difference would have been insignificant.