according to the Longman dictionary finely is an adverb with 3 basic meanings.
1. very thin; 2.closely and delicately; 3. very well , especially in a moral sense .e.g. I think he behaved finely.
I think he did the job fine would be more colloquial than finely...but I also agree that he did a fine job is by far more used....Iīve heard many times the sentence "he did fine" meaning "he did well" though and I canīt help but remember that americans say things like " he did the work real quick!"
and yes, BE means British English and AE American ,and instead of saying a "big" job itīd be better to say a "great"job since big would give the impression of something big in size rather than in quality.