Neil, I don't know how many public putting greens still exist in the UK, or how you would find out. I think the main problem is upkeep and vandalism. Also you need to pay someone to give out clubs and balls and collect green fees. There is a public pitch-and-putt in Wilmslow (just put The Carrs, Wilmslow into Google Earth and it will find it) and I played it a few times when the children were young, but some of the holes are crazy and maintenance is poor. There was rarely anybody else playing or anyone to collect your fee so you could make up your own holes which was great fun. The only time there was anyone to collect a fee was in Wimbledon fortnight and for a short period thereafter when the tennis courts were busy.
In fact there is a good par-3 course (Hawtree-designed) nearby at Adlington which has two courses, rather better maintained. One course is pitch and putt (longest hole ca 85 yards, shortest about 40), the other is more serious ranging from 110 yards to 240. If you put the postcode SK10 4NG into Google Earth it will find it (well, a little to the south-east, to be accurate). To the north-west of it is Woodford Aerodrome where Avro lancasters and Vulcans were built. To the south of runway 07 is Avro Golf Club, which I presume was started by employees but is, I believe, a provate club these days. It is a 9-hole course with 10 greens, and oodles of character. I've no idea who may have designed it.
Here's a challenge for you, Neil. What can you find out about Marbury Country Club (ca 1933-1939) just outside Northwich and Hurdsfield Golf Club (just north of Macclesfield) which I think also perished in the war and is now a factory estate? It, apparently, had a high reputation pre-war - and it had some good competition from Prestbury, Stockport, Wilmslow etc.