Update.....
I spent close to 2 hours on the phone to Apple tech support on Friday. After a good 30mins trouble shooting/various operations (All of which I had already tried) with the initial chap, I was then passed to a tech in Ireland (A very helpful soul). We tried creating a new user account and running messages via that, safe boot etc etc. He checked my Apple ID for any abnormalities (All checked out) and various other things until telling me he would do some investigative work and call me back. In just over an hour he returned the call and told me it was due to the logic board replacement and that he would need to request for my Apple ID to be "White-listed". This, he attempted to do there and then but for some reason couldn't process it. The call was finalised with him saying it would take 24hrs for this "White-listing" to take effect and that he would call me back with an update, however, in that time there could be a possibility that messages would start working again as normal. Either way, he would call me back. Think it is all to do with the macbook not pairing up my ID with the new hardware. So in theory, my ID needs resetting with the hardware (I think). I tried signing out and back in again this morning but there is still no joy.
One thing that was strange that I noticed was that when I go to Preferences in Messages - Details - then enter my Apple ID Password, View Account. Where it has the boxes to enter your Name and Password, the password boxes that have bullet points in there is one bullet point missing in my password length. I flagged this up with the tech and asked if there should be the same number of bullet points as characters in the password. He said yes. So I tried entering the password again, hit save and got red text reading "You must choose a password that you have not used in the past year". All very strange as I have always held the same password for my account and it works on other devices for messages, iCloud and everything related to Apple ID.
Oh and I also got an email after purchasing an album on iTunes (The first I've bought since logic board replacement) stating "Your Apple ID, XX@XX.com, was just used to purchase XX by Various Artists from the iTunes Store on a computer or device that had not previously been associated with that Apple ID."
Yeah, ok. I've only purchased on that for the last 5 years.
Anyway, I can only suggest you get onto tech support and hope they can sort it with this White list thing. I will post again with any updates. Hopefully this will be resolved for me today. I'm now on day 3 since the call.