My understanding is the following:
Mortgages are offered in the UK based on earnings (usually you'll be asked for proof of earnings from the previous 3-6 months). Multiply your joint annual income by 4.5 and that's usually the maximum figure you'll be offered. Depending on the amount you'll put down as an upfront 'deposit', you'll reduce the interest you're offered on a fixed term, which usually lasts between 2-5 years (at which point you can either re-negotiate your fixed rate, transfer to a different mortgage provider or automatically move onto a variable interest rate). I'm fairly certain you can't get asset-based mortgages unless you receive dividends from said assets, which you can provide in your proof of earnings.
You'll pay stamp duty, which you can read about here:
https://www.gov.uk/stamp-duty-land-tax You can use this calculator which is quite useful:
https://www.moneysavingexpert.com/mortgages/stamp-duty/ Being a landlord in the UK carries a few responsibilities and taxes/insurances.
https://www.gov.uk/private-rentingThere's also some good information on a buy to let mortgage here:
https://www.moneysupermarket.com/mortgages/buy-to-let/ They're more commonly referred to as an 'interest-only mortgage', as you only pay the interest.
My honest assessment is the headache of becoming a landlord from overseas will far outweigh any benefit in purchasing a buy to let property. Tenants have numerous rights in the UK that US citizens don't possess. You'll have to organise a company to manage your tenants presumably, which will eat up any profit. Property in the UK is a reasonably safe investment though. Our property market is more stable than the US. Where I live in Manchester, house prices usually double every decade. So buying isn't a horrible option - I personally just couldn't do with the stress of being a landlord, never mind being an overseas landlord!
Do you know Edinburgh well? Do you know which areas to buy in etc. I feel you'd be better moving and renting a property in the city and getting a feel for the place if not. You'll find which areas you prefer, and when the time comes to buy a property you'll be flexible.
If you have queries about mortgages in the UK, moneysavingexpert is a really useful website.
:)