In UK law there is no requirement for a company to provide a reference of any sort at all - but it is normal practice to reply to a suitably worded letter with at least the dates of employment, if nothing else.
You do have other ways of proving your dates of employment, as you should have payslips, for example, and perhaps your original offer letter.
If still employed I'd suggest having a few quiet words to find out if there is anyone else a new employer could contact. The HR function is the most obvious starting point.