NOTE that Bovington isn’t in London, it’s in Dorset, which is 125 miles SW of London.
Edinburgh to Bovington is about 450 miles. There is no way you can do this itinerary easily by bus/train, especially if you want to include Alnwick and Hadrian’s wall. ALSO be prepared for snow which can bring the country to a standstill...
450 miles may not seem a vast distance to you, but is going to take a full, day to drive - and yes you will need to drive. Doing Alnwick and Hadrian’s wall with add in extra miles and time. You will need to factor in at least one overnight!
Take the A1 to Alnwick for Alnwick gardens and then cut across country through Rothbury and Otterburn to pick up the A68 for Housesteads.
Is Alnwick Gardens worth seeing at that time of year? There will be few flowers around although there should be a few early bulbs. The trees will still be bare. This means you will be much more aware of the underlying structure of the gardens and the massive effort to create it. These pictures were taken in late March, but will give you an idea of what to expect.
Allow plenty of time for the gardens. They are huge and need a minimum of 2-3 hours.
Hadrian’s Wall - try and allow time to walk some of it. Make sure you have sturdy shoes as it could be muddy. BUT also bear in mind at that time of the year it could be covered with snow! Houseteads Fort is possibly the best place to head for. As well as the fort (don’t miss the latrines!) there is a lovely walk along the wall to the west - walk as far as legs and time allow. In your time frame, this is probably the best use of time.
This is going to take you a full day, so plan an overnight in say Hexham, Corbridge or Haltwhistle.
Next morning use the B6318 which follows the line of Hadrian’s Wall (you can see it on the skyline to your right) to Greenhead where you rejoin the A69. If it is snowing, use the A69 instead of the B6318. The B3618 is a roller coaster of a drive with a lot of blind dips which will quyickly get covered in snow.
From Carlisle, pick up the M6 and head south. This is still a 350ish mile drive, so you may want to make another overnight stop. If so, I would suggest Bath, which is a wonderful Georgian City (think Jane Austin) and also has the most amazing Roman baths (although I can’t find anything about opening hours in Jan/Feb).
Don’t forget, in winter there will be restricted daylight and you may not want to drive in the dark. Poor weather, especially fog will also slow down the journey.
The Tank Museum website says allow 4-5 hours for a visit so that is going to take up another day. Near by are the ruins of Corfe Castle, which must be some of the best in England. Even if it is closed, they are still impressive viewed from outside.
So you need to factor in 3 days for the trip. Lindisfarne, Bamburgh Castle, Chester’s Roman Fort, Jedburgh and Melrose Abbeys would add another 2-3 days to the trip!!!