Could someone please advise of different breeds which do not cast. I am
aware of Poodles, Bedlingtons and Maltese but any others would be really
helpful. Young child desperately wanting four-legged companion but health
issues rule out breeds which lose their coats.

a Bichon Frise is one which does not lose its coat - but they do need a lot of groming to be kept looking matt free - however much brushing ours gets there are still mats to be found somewhere (although his coat type is not correct which probably has something to do with it) but they do have lovely characters

What size dog did you want?

There's the Bichon frise and a few others.

One of my friends has a Miniature Schnauzer & he says this does not moult.

I have two miniature schnauzers and its true they do not shed at all but like all
dogs that don't they have to be groomed regularly. They are great with children
and love a bit of fun and will not go mad if you don't walk them miles every day.
As you can tell I think they're great!!!
Good luck in your search

Have a look at Havanese they are medium size and don't shed