Just an update on this.
I went ahead and downgraded our interim site that had been using our custom domain.
After this, I went into settings within our new full-sized website and selected ‘Custom Domain’
We have control over the domain which was registered with Google so our domain actually popped up in the field where you type your custom domain.
We applied that and snap! Our site had the new domain set.
So to answer my own question, if you have control of the domain and it was a domain you have already used on another version of the site in the same workplace, the time this might take to apply domain and publish site is probably less than 2 minutes.
One minute and 30 seconds of that was just double checking that you are in the correct site etc. etc.
Very easy to do but not having done it I really didn’t know what to expect.