The sums are beyond me but I doubt the moon would exert enough force on a large lathe to effect it directly.
But, if a long lathe was located close to the sea-shore, I can certainly believe the weight of sea-water moved by the tide would be plenty big enough to noticeably flex the ground the factory is built on.
Vosper's were based in Portsmouth which has a tidal range of about 4 metres. Guesstimating the Solent to be about 90sq km, that's 360 million tons moving about just outside the front door The tailstock and headstock of a long lathe close to the foreshore might well move at different rates, making it sensible to work only while the tide is on the turn and the ground is still.
The ground also moves itself due to tidal forces. Wikipedia mentions up to 380mm up/down and 50mm sideways, though 40mm would seem more usual. But inland, everything under a large lathe would move at the same time. Quite a few very big lathes are vertical. Dunno if that's to do with the tide, but its much easier to keep a column straight than a beam.
Dave