Hi,Had some success with other hospitals in London by sending letters enquiring after a relative to, say, The Archivist' or 'Personnel' at the relevant hospital.
You are fortunate, Katie. The National Hospital for Neurology & Neurosurgery (as it is now known) in Queen Square - note "Queen", not "Queens" - has a very well preserved archive. Much of their material, including staff records, has been digitised but is not available directly online. You can, however, request copies of documents via its website https://queensquare.org.uk/archives/about/archives