Surely a decent, strong shed that has double glazed windows and insulated walls and roof would do the job?
Running power from the house shouldn’t be too hard and you’ll need a concrete slab or similar to put the shed on.
You could always set it up with just a few lights etc and add more if needed, use desk lamps etc. Add an electric fire/ heater and you’re sorted.
Not sure how much a decent shed is these days but I’d be very surprised if the whole lot comes anywhere near £20k.
Just an example -
https://dunsterhouse.co.uk/garden-offices/dominator-insulated-garden-room