"you'd have formalised that arrangement with an indefinite visa or a British passport and/or kept hold of the evidence of when you arrived"
Or, you know, we could assume they are in the right and take a look at other records to determine the validity of that. Records like NI contributions, educational history, health records, electoral role, tax payments...
But no, far better to assume that they are illegal and ignore all other evidence to the contrary. Typical of May and Cronies.