There are two series of TR3Bs. The early version was TSF and the later one was TCF. Before I stated the differences, I checked some of the books in my library.
According to Piggott’s “Original Triumph TR2/3/3A” the TS series had basically a TR3A drive train. This would imply that they would have a non-synchro first gear. Piggott's book says that the TCF series had the basically a TR4 drive train including the full synchro gear box. The “Illustrated Triumph buyer’s guide” by Richard Newton agrees with Piggott about the engines, but states that both versions had full synchro gear boxes.
Not all of the TCF cars had the 2138cc engines. A few months ago I exchanged some emails with another TR3B owner who’s car had been ordered with a 1991cc engine.
My car, TCF5##L does not have the original engine, but I think its transmission is original. Its serial number starts with a TSF. This would tend to support the idea that at least some of the TSF cars had full synchro gear boxes.