I can't see the route being viable until the whole line is high speed. That might just about enable one return journey per day and a service frequency like the Eurostar Marseille service. London - Milan is probably similar. I can't see any UK government supporting Schengen, if we where to rejoin the EU at some point then a lot of political capital would be put into requesting some sort of associate membership of Schengen for UK (if Ireland agreed), to retain border controls.
Ireland effectively has a veto over UK membership of Schengen for as long as Northern Ireland is part of the UK. I think they would have joined if UK had pre migrant crisis but would have little interest now. In the event of a United Ireland either country could join seperately but the politics of that would be fraught.
Until such time that our country grows a pair and joins Schengen, there wont be any long distance (over 7hrs) trains from London.
Ireland effectively has a veto over UK membership of Schengen for as long as Northern Ireland is part of the UK. I think they would have joined if UK had pre migrant crisis but would have little interest now. In the event of a United Ireland either country could join seperately but the politics of that would be fraught.