Is it on a Linux server at the minute?
if it is I would get a copy of it along with the database and create a local copy (running XAMPP if you’re on Windows) that you can apply the new theme to and then when you’re happy with it push it to the live site, obviously this will require some configuration and changes to get working but would be the best way forward IMHO.
Failing that create a staging.yourdomain.co.uk sub domain and get a copy of the site up and running there to do your changes on.
just make sure to make a note of the files you edit (maybe use a child theme) so you don’t do a theme update and overwrite your changes and then have to go back and try to remember what you updated.