Doesn’t look you have a google ‘my business’ page setup, so that should be your first step. Will get you google maps search placements too. Create Facebook pages, Trustpilot pages etc too, IME it helps fill up the results page for brand searches with things relevant to your brand.

Getting your version higher than the US version I wouldn’t see as the primary goal. If they’re searching for you by name they already know who you are. You want to try and get ranked for e.g. driving lessons Luton. So go read a book and focus on local SEO.

Also if you can do that then searches for your name by UK customers will probably end up replacing the US version anyway as google sees you are increasingly relevant to driving schools.