Someone is threatening me with court action and has sent me a letter without an address or signature. Is this legally binding?
This sort of letter is not usually going to be legally binding in the sense that a contract is legally binding (whether or not the letter signed).
I'd need to see the letter in order to comment further.
However, one point which it may be useful for you to know is that under the rules of pre-action conduct that apply to claims in the English courts, a formal letter of claim should meet certain standards. For details, see:
A letter which did not identify the potential claimant would not usually meet these standards.