If your first stop is Glasgow, I’d definitely fly to Glasgow from Heathrow, especially as you are already at the airport. It should be much quicker than going into London and then catching a train (the train ride itself will be anything from four and a half to more than seven hours) At Heathrow, you’ll have to clear immigration and probably customs too (and if so, then check in your bag again) so allow plenty of time between flights. I’d suggest 3 hours although you may well not need so much time. It’s just possible that you can check your baggage all the way through to Glasgow if you are on the same airline and then clear customs there.
From Glasgow onwards as you head south, train or possibly coach (bus) would work since your journeys will be shorter. Train will usually be quicker than coach although the British rail system is rather Byzantine.
I like the Rome2Rio app for getting an idea of what the travel options are, although Google maps is fine too.