Can i pursue a cra (transunion) for holding incorrect information ?

[Ash17]

HI , this is a follow up to my original post where I was getting a mortgage rejected as Transunion were holding a CCJ against my credit file. EVEN though the first name was totally different and it was at an address I hadn’t lived at for over 10 years. I had to doggedly pursue them and even threaten them with legal action and report them to the ICO. They have also lied to me on several occasions (of which I have documented proof) . They have now said the “RTL are agreeing to remove this so we will update your credit file” when I contacted the RTL they clearly stated they had already responded to Transunion last month and its up to Transunion on how they interpret CCJs. The woman on the phone did say that if the name is totally different she would be surprised for them to attach it to my credit file but clearly stated it was up to each CRA on how they interpreted what was on the register. Both Experian and Equifax have not attached this to my file and never would as the CCJ is totally different to my first name but the Second name is the same.

Anyway now that I have an admission that “RTL have approved the removal of the CCJ", that they will remove it and RTL i don’t believe had anything to do with them attaching it to my file - I believe it should never have been there in the first place and I have lost out financially to the tune of around £10000 , am I able to pursue Transunion and what would be the best way to go about this?

Any help or suggestions would be most welcome.

Kind Regards