1. Some routers can be configured to function as a "Wireless Network Bridge", which will then connect a "wired" network to a "wireless" one (your Sky Router). If your Belkin router doesn't have this option, it can't connect wirelessly to another router.
2. There shouldn't be any reason why a computer can't have two network adapters connecting to different networks, i.e. your "wired" network by ethernet and your wireless adapter to the Sky router. Having said that, there can be a problem with Windows *VISTA* where it only allows Local Access with a network adapter. This doesn't happen with XP and has been fixed in Windows 7. Is this your problem?
Also, when connecting to two routers, the two networks will have to operate in different Subnets (IP address ranges). This shouldn't be a problem as the Belkin will use 192.168.2.x addresses and the Sky router probably uses 192.168.0.x or 192.168.1.x addresses.
Should I upgrade to Windows 10? 8 reasons why you should upgrade to Windows 10... and 2 why you…