AFAIK only generally once or twice per day for the Waterloo-Weymouth Quay boat trains (does anyone have any timings for them?), although I believe that for a short time in the early to mid 1980s (1984 and 1985 according to
http://www.disused-stations.org.uk/w/weymouth_quay/) there was a regular Bristol-Weymouth Quay DMU service at least in the main summer holiday season. Not sure how many times a day that ran, though.
The pattern of boat train workings over the years varied as the smaller ships were replaced by the Caesarea and Sarnia in the early 60s (they weren't built as ro-ro BTW; the first such to the Islands was the Caledonian Princess around 1975).
Services were transferred from Paddington to Waterloo in 1960, and at that time there was:
8.27 Waterloo - Quay 12.00 and 3.45 return for the day boats
7.45pm FO and 9.20pm SO returning at 5.55pm SO and 7.0am Sun.
I've ignored reliefs; there were no Sunday boats and no daily night boat. The Saturday day boats also had a Birmingham connection - WR stock of course, about 5 coaches (the Carriage Working Notice is confusing!).
Edit: I hadn't realised that a night boat from Southampton continued into the early 60s (9.15pm Waterloo); this returned as a day service, the boat train leaving the Docks at 5.35pm.
Once this ship was withdrawn, Caesarea and Sarnia worked day and night services from Weymouth, so the service
via Weymouth actually
improved.
From 1967 the workings (from memory) were:
0937 Waterloo for the 1330 boat, returning 1550 and forming
2008 Waterloo for the night boat, due back into Waterloo about 1007
except on Saturdays when it ran much earlier so the stock could form the 0937 out and release the second set for Exeter line duties.
The Bristol DMU ran at this time (not sure of exactly which years), alternatively a Southern National bus was used from and to Town for Bristol passengers (photos online - I googled Lodekka Weymouth Quay).