Changing site URLs

Hi,
I’ve built my site on my own hosting (eg. https://www.myhosting.co.uk) and have got the client to point their domain at this. Everything is working fine except the urls for links, images etc still show as https://www.myhosting.co.uk. Can I update the urls in general settings to the client’s domain? I’m a bit scared to do this as It’s caused me problems in the past.

Any help would be massively appreciated.