I have no experience of the B737 but I have relit a lot of different engines for trials purposes (and a few for real failures) so I make these comments:
While it may be possible to get high altitude relights the most reliable relights are to be had below 25,000ft.
It is possible to actually damage engines trying to light them too high if they go into surge and you do not turn the fuel off within a second or so.
So I would forget relighting drills and concentrate on gliding for range, sorting out ATC and deciding which airfield to aim for.
Because of your height you have ages before you will get down to 25k so things should be quite calm when you do eventually do a relight drill on one engine. If it fails to light I would stick to that engine until passing 20k when I would try to light the other one.
If neither lights by 15k concentrate on flying to your force landing diversion.
Once below 10k the non handling pilot should try more relights but the pilot flying should not be distracted from the force landing attempt.