To be clear, Romanian tailors are not running operations like Hong-Kong or Bangkok - i.e. short term turn-around destined to cater foreign travelers - it is mainly oriented towards local clientele, like in Southern Italy or Greece. The price difference is given by the relative value of work price in the general context of Romanian economy.
Time - of course they can do a rush job, but I, personally, cannot guarantee that. Three weeks is quite a good turnaround - for three fitting - if you compare with London or New York. Btw, you can find MTM operations in Western Europe that are comparable in price with Bucharest, if you add travel costs, etc. I am saying that mainly because you want to start with just a coat...
Fabrics - I am buying mostly over internet, there are at least three fabric shops in Bucharest - two will sell Italian "designer" and Zegna fabrics - one I suspect is dealing mostly in remnants bought from Italy, sold at quite a hefty price - the other one is selling Romanian made fabrics - the pure wool is ok, but only if you want a substantial classic fabric. Of course there are Brioni, Scabal, Isaia etc., but be prepared to pay a higher price than in US or even Western Europe.
So, if you did not made your mind, it is quite a lottery - in the sense that you have to come prepared with very specific demands and style,including your fabric, and having in mind the fact that you are dealing with honest artisans, more than with stylists. Probably they can do Tom Ford - never tried - but you have to be the stylist.