It's not something I'd personally recommend, seems like it'd be a waste of money and a hassle to do. Don't forget w a US car you'll be on the wrong side of the car for the road here (maybe not critical, but may not be comfortable/convenient). I'd imagine selling your car and purchasing a secondhand car (or even a new car depending on make/model) here would be cheaper and more straightforward. Occasionally see vet students selling their old cars too.
Keep in mind too -
You can only use your USA license for 1 year from arrival in the UK, then you must get a UK license (theory and practical test).
All cars must have insurance, road tax, and an MOT (iirc MOT is yearly for any car over 3yo); all must be renewed annually.
Depending on where you live, you may need to pay parking charges (certainly anywhere within the city, unless you find a house with a private drive).
Petrol generally costs more than the US and is charged per litre not per gallon.
Think too about whether you'll need a car or not, and when you may need it - it may not be necessary right away. I can't speak for glasgow and rvc but Edinburgh's transport links are good, regular busses out to the school if you live in town and busses from elsewhere if you decide to live closer to school.
If expenses are of any concern, i can say that especially with the school subsidy, busses are cheaper than driving when taking all that into account.
Only thing I could imagine is if you cannot drive stick, you'll have less choice trying to find an automatic (at least for less £ than an otherwise equivalent stick shift), though if you fancy taking a few lessons to prep for the eventual license tests, could learn stick shift then.
https://www.gov.uk/browse/driving for some more information on things like MOTs etc
https://www.gov.uk/importing-vehicles-into-the-uk/overview info on importing cars
https://www.gov.uk/government/organisations/driver-and-vehicle-licensing-agency the DVLA