I think that it could be possible to improve things, however only if the airlines were onside. That would probably require such a significant modal shift that there were few people wanting to use the connecting flights.
For instance, they already have to process your luggage without it going to the standard luggage pick ups, as such it could be possible to arrange for your stuff to be transferred to your HSR service with you "collecting" it as you board the train but with a member of staff loading for you as an optional extra.
Given that there's probably something like a dozen people an hour, that's probably achievable to do.
It would probably need to head to OOC separately to you, but given that you have to clear immigration and customs whilst you're luggage doesn't (although it could need to have checks done to it to limit it being a useful smuggling route, or you would have to talk to a customs officer as part of the pick up process at the train station) chances are it could get there ahead of you.
I agree it's not insurmountable, but it's certainly not easy. Is there anywhere else where airline to HSR interlining is frequently done for journeys this long?
As for numbers, the majority of domestic passengers at Heathrow are connecting, so it's more than a few dozen in total. It's also nowhere near evenly distributed across the day IME.
I think it's unlikely HS rail could ever supplant interlining at Heathrow. Even with new infastructure all the way and a dedicated Scotland-Heathrow train (neither likely to happen) a 2hr train plus a 1hr check-in can't compete with a 1hr check-in plus a <1hr flight. Experience on the Continent suggests that a 3hr journey time would capture the majority of the domestic air market though, and with increased security in recent years that figure may have crept up a bit. The latest HS2 forecast is 3hr45min.
So, from where I live in Glasgow, connecting on to a long-haul flight from T5:
0:15 - taxi to airport
1:30 - at Glasgow airport (and this is pretty generous! especially in the mornings I often have a sit-down meal at Spoons in that, though it provides a decent margin in case of extreme traffic to the airport)
1:30 - GLA-LHR
1:00 - MCT onto connecting flight
Total: 4:15.
Compared with by train, assuming a hypothetical 2hr train:
0:15 - taxi to Central (depending on how much luggage I have, I might actually get a suburban train there, but that's end-to-end slower, so let's keep it fair versus the plane)
0:30 - at Central (note the minimum connection time is 0:15, so this is arguably generous)
2:00 - Central to Heathrow Airport Station
0:15 - Heathrow Airport Station to T5 (I think this is roughly right for the current T2&3 station?)
1:30 - at LHR before boarding (note the 1:00 cut-off for luggage check-in for long-haul, so we can't safely reduce this without ending up with little slack)
Total: 4:30. OK, so if we cut it down to 15 minutes at Central then we're even.
Onto a long-haul flight, even a two hour journey from Central to Heathrow is only matching the time of flying from GLA.