There was a series of reprints of special editions of "The Shipbuilder" published by Patrick Stephens in the 1970s. Thse editions were originally published to commerate the launching of an especially notable ship or sister ships.
Volume two detailed the building of the 'Lusitania' and 'Mauretania'.
There is a full page broadside view of the 'Lusitania' on the stocks with the propellers installed. The caption: "Lusitania" on the Stocks ready for launching".
There is also another view from the stern with the caption "Stern View of the "Lusitania" on Stocks, showing the Propellers, &c.
The complicated bit in this area of shipbuilding is fitting the propeller shaft, not the propeller. I will try to describe the process in simple terms
The shaft is made from very special steel. It arrives at the shipyard slightly over diameter and is turned down to the precise size in a very large lathe.
In the meanwhile the shaft brackets and their bearings, the stern tube bearing and the shaft tunnel bearings are all installed in the ship. The centre line of the shaft is then established (By laser today, by measurement and a wire in the past) A shaft-boring machine is then assembled at the stern of the ship. This is a shipyard specific tool, consisting of an electric motor and support brackets attached to the end of a long shaft, which is inserted into the precise centre of the line of bearings where the propeller shaft will sit. A sliding cutting head is attached and the inboard end of the cutting shaft supported. The fitters then bore out each of the bearings along the line of the propeller shaft. The newly machined shaft is then slid into position and secured.
The installation of the propeller shafts is probably the most highly skilled machining activity routinely undertaken by shipyard fitters. By contrast fitting the propellers is a muscle job. In answer to your question Ken, the propellers are usually fitted prior to launch. It is quicker as there is more room on the building berth than there is in a dry dock; it avoids corrosion to the exposed shaft and it enables the builder to run basin machinery trials at the fitting out berth, prior to pre sea trial dry-docking.