I have already referred (in my article on Epiq’s costs seminar) to Murray & Anor v Neil Dowlman Architecture. There is a good commentary on that on Outlaw.com called Cost budget revisions to fix mistakes unlikely to be allowed, says judge which you may find helpful.
I have mentioned also Dominic Regan’s article The end of late chopping and changing on the same subject.
Distinguish between amending a budget because the facts have changed and getting the budget wrong in the first place.