Re: routers and modems. A modem modulates the in/out digital signal. A router 'routes' the signal between networks or devices on the networks. In the old days you would get a modem and a router separately; nowadays the two are combined.
So you have two possibilities:
1. Phone box on the wall - micro-filter/splitter plugged in - flat plug lead to phone - square plug lead to modem - another lead to a router.

2. Phone box on the wall - micro-filter/splitter plugged in - flat plug lead to phone - square plug lead to router/modem.

The wiring described in my earlier post is probably already the case, see the pics, but with a possible Ethernet lead to your Sky box. With the Sky box turned off that can be left as-is. If you have the option with the modem and separate router (pic 1.) it is time for TT to give you an upgrade.
The reason for disabling the wireless connection and enabling the local area connection (Ethernet) is so that your connection is by wire. This is faster, truer, more secure and less subject to interference than wireless.
The whole thing should take no more than 30 mins.