Timeline, using the official designations,
Supermarine, October 1941 first Vc, February 1942 last Vb, June 1942 first F.IX, September 1942 last Vc, November 1942 first F.VIII, February 1943 first LF.IX, May 1943 first LF.VIII, June 1943 last mark IX.
Vickers, June 1942 first Vc (1 aircraft, next 5 in August), December 1942 last Vb, February 1943 first F.IX, August 1943 first LF.IX (ignoring production of 1 in April and 1 in June) October 1943 last Vc and F.IX
Westland, first Vc April 1942, last Vb June 1942.
The RAF aircraft census starts using the Spitfire F, HF and LF designations in May 1943.
The Ministry of Aircraft Production put out monthly production reports, from March 1942 onwards at least. No doubt they do not exactly match to when the RAF made designation changes to Spitfires, but they give an idea of the timetable.
First mention of F designation, F.VB in December 1942, the last production. The F designation does not appear again until March 1943, when all Spitfires in production receive it, Vc, VII, VIII, IX and XII. In April 1943 the HF and LF designations appear. F.Vc, F.VII, HF.VII, F.VIII, LF.VIII, F.IX, LF.IX and F.XII. The F and HF VII appear in May but then become F.VII in June while April and May VII production is given as a single figure, not broken down into F and HF.
Before December 1942 the reconnaissance version Spitfires were designated P.R.U. III and IV. In December 1942 the PR designation is used, the first PR.IX were built in November 1942 but the MAP reports miss them.
Examples from other types,
In March 1942 the Beaufighters have c or f suffixes to mark numbers. The TF.X appears in December.
In March 1942 the Mosquito are F.II and T.III (in June 1942 the first mark VI is an F.VI). The mark IV becomes B.IV in September, the same month as the PR.VIII appears. In January 1943 the NF.XV and FB.VI appear while the F.II designation is still in use.
In May 1942 the Wellington Ic are split between Ic and Torpedo Ic also there are Torpedo and Leigh Light VIII. In April 1943 the GR designation is first used.
In March 1943 the Warwick I becomes the B.I.
As a warning about the limits of the reports, in October 1943 the Halifax II becomes the B.II but the mark III and V do not use the B designation.
And so on.