Hi Alchemy,
Sorry to hear that you are being messed about like this, but despair not, the law is on your side.
Disclosure: I have a small firm building garden rooms and landscaping now, but I have been in the building trade for 28 years, I am not touting to get involved in the job, nor do I have a specific roofing company to recommend, I have been hiring tradesmen, and firing them if they weren't up to snuff, for many years so hopefully I can help you fight your corner.
Any information I give you will be for your information and is based on my own experience You must decide what action to take, and I cannot be liable for the results, I am just trying to help, I am not a Lawyer, I am just trying to say what I would do in your shoes.
I hate these cowboys too as they give us all a bad name.
You have the right to expect good service from anyone purporting to be skilled at their trade.
https://www.citizensadvice.org.uk
“The Supply of Goods and Services Act 1982 says that reasonable care and skill must be used while working."
If you gave the go-ahead on or after 1 October 2015:
“The Consumer Rights Act 2015 says that reasonable care and skill must be used while working.
So there are now two acts in law protecting you from poor treatment at the hands of these chancers.
All communications with them should be in writing from now on
so here is how I would go about it
Important: Get my shot in without delay; I would write an email and registered letter to the roofer and I would make these points
"I will not be paying your invoice for the following reasons."
"In my opinion, you did not use reasonable care and skill when you carried out this work."
Your invoice at £3100 is far beyond the amount that I expected for three days work and I will not pay the amount demanded by you until I am fully satisfied that the charges are reasonable.
" Stop doing any work to my property. immediately.”
(just in case he rocks up to "Fix" the problem) this is effectively firing him. I would need this point to be made to avoid conflict and if it goes to county court I will need to say that I informed him to stop, and show it in writing.
"I will now need to have your work professionally assessed and where necessary I will rectify the work and invoice you for all costs including my costs and any professional and material costs relating to correcting the work done by you or your anyone working for you and for work to correct any damage which may have been caused to my property through your careless work practices."
I would Consider sending by courier, and email. That being done will allow for time to get the work that he has done assessed and allow time to process all that has happened I know it is not your daily thing so you might be learning a bit about roofing and roofers but knowledge is power and they have you on the backfoot until you empower yourself.
Ps: This action would mean that I would be officially "In dispute" with the tradesman, but if I did nothing this then eventually I would just be liable for the invoice as it is.
Hope this is helpful.