I see enough demand for services from Birmingham, Manchester and Leeds HS stations to stations on the continent. There may be enough to justify the HS1-HS2 link in it's currently proposed form (ie single track), but if it proves too difficult, I can understand why it wouldn't be built as double track.
Taking the current London services as a starting point (assuming HSTEd is correct and there are more trains to Paris than Brussels):
- London - Paris 1 per hour
- London - Brussels 0.5 per hour
This will be increased next year (?) by additional services. Say (for the sake of argument) that it ends up as 0.5 trains per hour. So after next year, you would have:
- London - Paris 1 per hour
- London - Brussels 0.5 per hour
- London - Amsterdam/Frankfurt 0.5 per hour
I can see the "provincial" stations supplying about half that demand on the HS1-HS2 link each, which would give:
- Birmingham - Paris 0.5 per hour
- Birmingham - Brussels 0.25 per hour
- Birmingham - Amsterdam/Frankfurt 0.25 per hour
- Manchester - Paris 0.5 per hour
- Manchester - Brussels 0.25 per hour
- Manchester - Amsterdam/Frankfurt 0.25 per hour
- Leeds - Paris 0.5 per hour
- Leeds - Brussels 0.25 per hour
- Leeds - Amsterdam/Frankfurt 0.25 per hour
By my reckoning, that's 1 train per hour (per direction) on the link after HS2 Phase 1 is complete, rising to 3 trains per hour on completion of the Leeds and Manchester links. I could easily imagine that the 1-track line could take that. (As an aside, this would mean 5 trains per hour on HS1.)
Of course, this also ignores the possibility of providing international platforms on non-HS2 stations linking up to HS2 (or even HS1). At the moment people are talking about an interchange at Old Oak Common from the GWL to HS1/2. Pretty soon there would be pressure for direct access from GWL (Bristol/Cardiff to Paris, anyone?). The GWL should be electirifed by then, although there may be other gauging issues.