Repair shops I have used always furnished estimates.
Wow! That’s puzzling to me…
a misfire that causes a flashing check engine light, yet no DTCs (diagnostic trouble codes).
Cost? I’m not sure what an injector for a German car goes for in G.B. Add to the parts price some labor for diagnosis and replacement.
Is this the shop that guessed at the coil and replaced that?
CSA
I think the coil was damaged when I first took it in as that was what was showing up on the codes and the engine light was on permanently. Because the car was ok after that (engine obviously warm from test driving etc), then this is at the same garage but they have taken it to someone who is a BMW specialist and it took him pretty much the full day to find the problem. He said he would call when he has a price for the part/labour.
Sounds good. Diagnosing and replacing a fuel injector, even on a BMW, shouldn’t break the bank. If they’ve got it fixed it’ll be well worth it to you and you’ll love the results. Good luck! 
CSA
Not applying directly to your particular engine, but for the BMW 3L 6 banger N51 engine, from what I see the injectors cost about $160 ea, and about 3 hours labor . I’m assuming that’s the labor to replace one injector. So you’d be looking at a little less than $500 parts and labor, presuming the part price and labor is roughly the same your car to a similar American version.
It’s possible the injector’s electrical connector got damaged during the coil replacement job. Unlikely to have happened that way tho. Still, injectors aren’t a common failure item.