Just a picture of the bottoms in case it helps - the insides are non-stick (the small one) and possibly white enamel (the larger one).

Any ideas gratefully received because we want to buy some more pans like these!

It appears that the bottoms are not flat so only a small portion of the pan would actually come in contact with the induction base. Right?

No they are flat, it's the shadow and perspective of the lens that might make them look round.

The closest I've found are Stellar cast iron pans, they're enamelled but no non-stick ones - link below:

Here is a link that might be useful: Stellar Cast Iron Cookware