Isle of Sky and Hadrian's Wall -- 10 days

If that is all you want to see, it's doable, as you could allot 4 days to Skye and 3-4 days to the Hadrian's Wall area (I would recommend basing yourself in Haltwhistle). It would be a lot of driving, though as you're going in two opposite directions from Glasgow.

Important question. When are you traveling? Lodging on Skye needs to booked far in advance to avoid disappointment.

It is doable as long as you are prepared to only have a couple of days for each place. You couldn't have picked two places much further apart. Skye to Hadrians Wall by car is going to take AT LEAST 9 hours and that is without any stops.

Don't forget, if off a Transatlantic fligh, to add in a day to recover from jet lag. Getting to Skye will take best part of a day from Glasgow. Travelling from Skye to Hadrian's Wall will take at least another full day and you may want to plan an overnight stop somewhere between the two. Hadrian's wall to Glasgow is going to be about three hours drive.

What time are you flying out of Glasgow and will you need to drice to Glasgow on your last day for an overnight before the flight?

I agree that jet lag and time will be a factor. One possibility would be to take the train from Glasgow to Newcastle, which would give you a chance to recover a bit. Then rent a car in Newcastle, drive to Haltwhistle and explore Hadrian's Wall for a few days with your rental car.

After that, return the car to Newcastle, and take the train back to Glasgow. From there, rent a car and drive to Skye. That will break it up and make it more doable, in my opinion. The drive to Portree from Glasgow would be around 5-6 hours, but you can break it up a bit with stops at various scenic locations.

Actually, after rereading your post, it sounds like you are planning a few days in Glasgow before you head out to either Skye or Hadrian's Wall. So jet lag should not be a problem. But I still think that taking the train and back for the Hadrian's Wall portion would be easier than trying to drive all over the place.

Glasgow to Newcastle by train is over an hour longer than taking a train to Carlisle and renting a car there, although there is a half hourly train service from Carlisle to Haltwhistle and the AD122 bus service on the wall with other supplementary buses on parts of the wall.
Train Glasgow to Haltwhistle can be as much as an hour faster than driving it (depending on connections) and you won't have even reached Newcastle in that time.
There is also car rental available in Hexham.

How much of Hadrian’s wall do you want to see? There are day trips out of Glasgow that might be easier than renting a car and doing it yourself. Then you could just stay put in Glasgow, and rent the car when you leave for Skye.

isn31c, thanks! I forgot all about Carlisle!

I agree the easiest way to see Hadrian's Wall from Glasgow is to get the train to Carlisle. In my opinion the best way to see the wall is to walk it... you can walk east from Carlisle tho a day's walk will only get you to some small pieces of the wall. Or since you are planning to have a car- basically any part of the wall between Gilsland and Chollerford will be amazing to walk, and you can drive there. Or drive and then take a bus back to your car. Good luck with your planning.

Peter Carney is a fabulous guide for Hadrian's wall. Can provide local transport as well. Here is his website.
https://www.hadrianswall-walk.com/