Although you couldn't just take the Alstom double deckers and use them on HS2 as it would have 1100mm platforms instead of the 550mm or 760mm ones in Europe, I don't see it's necessarily a barrier to having double deck trains on HS2. If your entrance was at 1100mm then you'll have shorter stairs to the top deck and longer ones to the bottom. The accessible carriage probably remains as single-deck to avoid massive ramps internally.
The larger barrier would be with the cutbacks to HS2, there won't be enough captive services to make it worth ordering a microfleet of European gauge trains.
From this video:
Here's the standard door lobby with steps down to the lower seating area and stairway to the left to the upper level. Note there is no through gangway at the lower level. The toilet and stairwell block access at the door end and there is a blank bulkhead at the opposite end of the lower saloon.

Note how the level walkway well of the upper deck drops between the entrance doors along the centre line of the car, with the steps to lower saloon descending clear of it. The entrance height is a little higher than the French 550mm standard platform height, sufficient to be compliant with EU maximum vertical step requirements at 760mm platfrom while not achieving easy level boarding anywhere, except on the special accessible coach, which as NCT explains, has active suspension, which can also change the door height to more perfectly match 550mm platforms for unassisted wheelchair use and a turntable lift in the lobby to access the lower deck.
So another new train design kicking the level boarding can down the line for another generation, though I can see its a difficult area of design with operators wanting to be able to run all over Europe with its varying platform heights. While their new Avril design has other issues and problems, at least Talgo defined the gold standard on this matter with fully level 760mm boarding at all passenger doors and a level train floor throughout. That potentially limits the range of their trains, though the company also claim to be able to create similar 550mm variants, with their special split axle running gear allowing a low walkway to pass between the wheels. It seems very difficult to design a train with level boarding that can do both standard platform heights effectively and legally. Stadler has managed the best compromise so far in their 'Smile' with multiple doors at each height throughout, at a cost of internal ramps and of course is only a single deck train.
I considered the Eurostar/Thalys service group and all its routes and stations would be good candidates to become a fully 760mm platform railway in the shift to LB, seeing as Belgium, North Germany, the Netherlands, and UK international platfroms on HS1 already follow that standard.