Is this the correct setup and does this setup enable my Internet connections to use the DNS settings on the Sky Router?
If everything is working then your setup is fine. It's only the Sky router that can do name-resolution lookups.
At the start of your post you said that the two routers were on different subnets which will never work. The second sentence of that paragraph seems to indicate that you changed the Netgear's private IP address, and if that's what you did, then that's good.