Hi, to the best of my knowledge there's no option for earning miles on UK trains to either Dublin or Edinburgh. It's also expensive-ish, but you should be able to find a flight on one of the EU Skyteam carriers via Amsterdam or Paris.

If you have the time, I'd take a train from London to Edinburgh; there's lots to see en route as well as when you arrive in Edinburgh. Aer Lingus have a flight from Edinburgh to Dublin at around $100 (source Travelocity) and there are several options (cheapest is British Midland) for flying Dublin to London. Unfortunately, neither of these are Skyteam members.

Hope that helps ...