The primary restriction on what can run on HS1 is the signalling system- remember, the 390s (and other high speed trains) are limited to 125mph thanks to the assessment of the safe operation limits of track side signalling. HS1 uses French designed in cab signalling (that DB will have to fit to any ICE units that they want to run through) that's fitted to the 373s and 395s.
There'd be little point in fitting 390s with the TVM equipment- where would they run on HS1? Ashford shuttles? The tilting kit would be completely wasted as well- HS1 is designed to be comfortable at speeds well in excess of 140mph without needing tilt, and I'd imagine that the gearing of the 390s won't allow them to go much more than 140mph (their maximum design speed), let alone the, what, 400mph that would make tilting worthwhile on the long banked corners?