Here in Canada we are somewhat between the US and UK, which is not surprising. There is more use of riding carts than the UK but push carts are commonplace.
I have played many of the most exclusive courses in the country and pretty much all of them have push carts. The only ones that I can think of that may not allow them are ultra-private enclaves like Redtail or Memphremagog.
At the most exclusive old-money private clubs in Toronto, like Toronto GC and Rosedale, the default is to set you up on a push cart. At my club, Scarboro, we probably have about 1/3 cart riders, 1/3 trolleys and 1/3 carry. Also popular are the motorized walking carts which I think are a very good compromise. My club has lots of elevation changes, including the approach to the 18th green is very steeply uphill, and it is hard for many people to push the carts up the hill.
I think we are much more pragmatic here than in some places in the US that have practices that are very hard to change. Caddies are fairly rare in Canada other than a few places like Hamilton G&CC and the Cabot Links resort. Cabot Links also provides trolleys and they are quite widely used.