Security infrastructure aside, it does sound like a Eurostar link would be technically feasible in the e300 units, certainly up to Birmingham and the North of England, but they would go round London. The journey time would be too long from Scotland though. Though it is unlikely any station will be retrofitted for border controls with this for a set of units that are going to be scrapped.
Surely some of the new stations on HS2 would be wise to have this infrastructure in place, but would a European loading gauge train be able to go from HS1 to HS2 without a cross-London link?