Sawdust,
This just one isolated personal opinion:
Flat sawn wood can be used for back and sides but this is usually 'piece dependent'. When we speak of flat sawn, we are not speaking of flat grained. The grain of most flat sawn pieces is usually in a shallow 'u' or an inverted shallow 'u' configuration, depending on the builder's preference for the face surface. When this configuration is perfectly symmetrical, some very attractive patterns can result, but when it is not symmetrical, the resultant patterns (in figured woods) can prove difficult to book match and are best utilised as one piece.
Adequately braced backs are rarely problematic, and the bending of sides can go well if the piece has little run out, but is prone to diagonal splits if significant run out is present. Vertical side 'splints' around the levels of the upper and lower bouts are good insurance. The species of wood is a big factor.... finer grained varieties usually behave better than the very coarse grained.
Some (typically high end) luthiers use quarter sawn wood exclusively. They see flat sawn wood as risky from both the stability point of view and the fact that flat sawn pieces are often associated with instruments built to a budget. This is not even considering the endless discussion and arguments of sound characteristics from different wood types and cuts.
Historically, some very fine looking and sounding stringed instruments have been built from flat sawn lumber.
Personally, I have not made an Ukulele from flat sawn stock (and have no plans to), but I imagine that it would not be as risky a proposition as with larger instruments.
I would use quarter sawn stock for all soundboards and bracing.
This has always been a contentious issue and this thread will likely perpetuate that theme.
I suggest that you go with the majority.