Hi Caroline -
I think the answer to your question is that it is not an efficient use of your time to detour from Scotland into England to see Hadrians Wall. Plus there's very little to see of the wall proper in and around Newcastle. To see it at its best you need to see it crossing the wilder, more exposed parts in the centre of the country, around Housesteads and near Vindolanda for instance. Traditionally in summer there's a bus service - the AD122! - that runs the length of the wall, but the service timetable won't be announced until the new year I think. (Nothing to stop you checking in the interim).
Of course it's your trip and your call, but I think Mike's suggestion about the Antonine Wall is a better alternative and use of your time, and there's plenty of good stuff to occupy you in Scotland and the Highlands instead of detouring in the opposite direction to see Hadrians Wall.
We have a saying about these sort of trips - "you can't do it all, we'll save it for next time" which gives us an incentive to return! So maybe Hadrians Wall is a trip for another time - walk the entire length! - as far as I know, it's going to be around for a while yet!
Whatever you decide, have a great trip!