>I wonder if the term "baby sitter" has become more or less universal. Isn't "nanny" the preferred term in the UK, or is that reserved for an ongoing arrangement?

We'd only use a baby sitter for an evening or so, even if it were a regular evening, usually a teenager, family member or friend.
A nanny would look after a child in your own home (like Mary Poppins) and you can still get Norland Nannies with an updated uniform, although most don't wear uniforms.
A child-minder would be registered with the local authority to look after children in their own home
A nursery nurse or worker would look after children in a nursery (babies up) or a pre-school nursery (2/3 up).
I suppose the people who work in playgroups (organisations registered with the pre-school playgroup association) are called playgroup workers or something like that.