While we're on the subject of violet/purple-coloured flowers in lawns,
I've
got another one: it has just started flowering, slightly fleshy, rounded
leaves and a short "loo-brush-shaped" head. Any ideas?

This first comes to mind.
http://www.tiscali.co.uk/reference/e..._selfheal.html

Yes, it is self-heal, thanks - and very pretty it is too - will definitely
stay in the lawn although it does seem to be growing at a rate of knots.