I am a bit starved of info here.. having no idea of what kind of broadband Sky gives you.
Please confirm the internet works fine when using the new router.. ?? Is this a fibre service or some sort of HFC cable or FTTN type.
If it is only the TC that is failing but other clients in the network can connect to the Sky Q router and get internet. Then you could have a number of different issues. The easiest way is to give us a screenshot from the airport utility of the TC.
A hard reset should be sufficient but if you run into trouble I would use a full factory reset. This works somewhat differently.
Hold in reset button without power.. apply power without releasing the reset and keep holding for 10sec .. the front led will flash rapidly.
Then run through the setup wizard again. TC WAN port plugged into the new router with ethernet cable. It will have to go into bridge mode. The automatic setup usually figures it out.
Use all short names, no spaces and pure alphanumeric characters. Passwords same rules but can be longer.
Do not reuse the old configuration.. no files are lost doing this.
No internet connection it is possible the new router and the TC don't like each other..