I don't think they're really gambling. As for the approval process: to my understanding, the emergency process serves to push approval faster so it can be administered quicker to a narrower population group. For example, to my knowledge, not enough is known yet about the effects on children and pregnant woman for the Pfizer/BioNTech one. Once that comes out, the UK will have to go through a relatively large approval process again, where apparently the more fulsome EU process performed by EMA means they can add further population groups to the approval more easily. Or something along those lines; I might be getting facts wrong, but the overall idea is correct. Pros and cons - but so administering the vaccine to the rest of the population isn't a gamble.
As for extending the wait until the second dose: that's done in other places as well. (Did QC start doing it or were they just strongly considering it?) The idea is to quickly get some immunity in more people, to help flatten the curve in a situation that's getting seriously out of hand. A full and proper campaign can follow later.
I suppose a proper discussion of this stuff is rather for the COVID-19 thread; but my point is that these are all legitimate, practical considerations, which every country can make. None of it is in any way related to Brexit. (Apart from that, as suggested above, there may be more peer pressure in the EU to wait for the EMA's process rather than using a local emergency process. On the other hand, waiting for EMA means not having to use resources at home for vaccine approvals, meaning that those people can focus on other things. Again: pros and cons; and nothing to do with Brexit.)