Mears of Walsall will copy your suit. I needed a new hacking jacket in a hurry two years ago but I've never been happy self-measuring, so I sent them a sports jacket I had had made. Their copy was perfect in every respect and I am happy to recommend them.
Some time googling will bring to light many other companies that will do the same thing.
I know nothing about Mears other than what the website displays, which is an MTM program.
Not bespoke. Based on the heavy, big shouldered, British Country Wear they show, I think
you would be crazy to expect that they could recreate an Oxxford soft-shouldered typically
American jacket. Your best option is to deal with Oxxford or go to Savile row to one of the
tailors known for "soft" clothing like A&S or Steed. Naples is another option