The first 77 Spitfires had the two bladed fixed wooden airscrew. After this the MkI was fitted with a de Havilland, three bladed,two pitch,metal propeller. So you are correct to say they did not initially have a constant speed unit,though some did have the Rotol CSU fitted. Rotol was a joint venture between Rolls Royce and The Bristol Aircraft Company.
Back in 1935, following the success of their own Comet racer, de Havilland acquired a licence from Hamilton Standard in the United States for a variable-pitch propeller mechanism.
As of 24/6/40 de Havilland started converting all Spitfires,Hurricanes and Defiants in service to accept a constant speed unit.
De Havilland manufactured the relevant parts,including those to adapt the Merlin engine to drive the hydraulic pump. They did not need to change the blades but did have to increase their pitch range. The time allowed to convert a squadron was ten days and in less than a month,by 20/7/40,all Spitfire squadrons had been converted.
On 16/8/40 every Spitfire and Hurricane in service,over 1000 aircraft,had been converted.
The spinners fitted to all these aircraft would have been familiar to their German adversaries. They were VDM spinners,manufactured by Constant-Speed Airscrews Ltd in Warwick. They had bought the licence before the war.
Cheers
Steve