Malcolm Storey has queried his database to count the UK species that Linnaeus named. This yields a total of 3,799 species.
It’s only indicative of what the great naturalists of the past would have seen in their lifetimes, for the following reasons:
1. He probably didn’t see all 3,799 species alive and in the wild.
2. He may only have seen a small proportion of those actually in Britain, which I think he only visited once for a few months.
3. There’s some room for discrepancies in Malcolm’s database query, e.g. due to him being variously credited as ‘Linnaeus’, ‘Linn.’, ‘Linné’ or plain ‘L.’.