I'm sorry but I think you're trying to pack far too much into what is a relatively short visit. This is compounded by the fact that despite your trip being focused on Scotland (plus alittle in the north of England), you're flying to/from London which is around 400 miles away, thus using up a fair bit of your valuable time sitting on a train travelling from your arrival point to where you're actually going!
I'd definately look to fly straight to scotland, or if youre flights to London are set already, at least book a connecting flight on BA to Glasgow. You're arriving off a long overnight flight tired and likely jetlagged, the last thing you want to do is fight your way for at least 60-90mins to the other side of London on the underground with your bags in rush hour conditions of packed tube trains/stations, followed by another 4.5-5 hours of not particularly Scenic train journey. Hiring a car at Heathrow and driving up is infinitely worse than the train!
I realise that there's perhaps a cost saving on paper from flying to London, but Glasgow and Edinburgh (and to an extent Newcastle) are reasonably sized airports with direct flights to the USA and a wealth of connecting opportunities, so there should be some kind of reasonable routing and fare available. If you've not booked your flight already then I would look to fly directly to and from Scotland. Newcastle isn't too far to drive a back to Glasgow, so you could fly in and out of Glasgow - Icelandair often offer good deals from your neck of the woods. If you prefer to fly into Glasgow and out of Newcastle then you should be able to get an open jaw type fare from one of the airlines that serves both airports - off the top of my head that would include Air France/KLM, Aer Lingus and British Airways.
If you have booked the flights into London already then it may be worth contacting the airline and asking if you can add connecting flights with BA up to Glasgow and down from Newcastle onto your existing itinerary. If not then I would book a seperate connecting flight with BA up to Glasgow. Because it's not on the same ticket you will still need about 3 or 4 hours between your arrival time and departure of the flight up to Glasgow because you need to get your cases and check in again. OK this is still a bit of waiting around at Heathrow but it's still easier and quicker on the train.
If none of the above is possible and you must take the train then 4 hours between arrival and train departure time is an absolute minimum. You can have headwinds, congestion/stacking at LHR, slow baggage claim/immigration, delays on London transport (which can be Common, particularly at rush hour) - quite quickly you're rushing to catch your train. Obviously the train journey from Newcastle to London is a bit shorter and not after a long-haul flight, but it's still will probably easier to fly from there to Heathrow.
Notwithstanding the transport issues you're itinerary is too much IMO and you'll find yourself travelling/driving nonstop and not doing many places justice - You've no time to see anything particular in Glasgow, day 2 is really far too much and needs cuts, days 3 and 5 are busy too. One day in Edinburgh is not enough either. Bear in mind many roads in rural Scotland are narrow and twisty and it'll often take a long time to travel between two places that look close on a map.
Sorting out your transport issues might help a bit but to be honest I would look at cutting out a fair few things as well. Unless your heart is set on it I would cut the north of England and focus on Scotland. Driving all the way to Durham is a bit of a detour - it's a nice town but I don't know if I'd go all that way just for that. York is way too much matching top everything else!