A few points.
You do have rights in this situation but sadly you have missed a few elements that will prevent you progressing the issue further
You have to agree a limit to the costs at the start.
As long as you are specific about the type of work you want done then you don’t have to pay for unauthorised work. This is covered by the Consumers Act.
You made sure you were clear about what you want from the start but missed setting the limits of costs (which is required by the act to protect yourself).
the problem is in case law it’s been clarified as
“- at a reasonable charge, if no price is agreed beforehand”
Depends if you see the charge as resonable.
If they estimated the costs again this would help.
Sounds like it may be one of those “lessons learnt” situations.