You need to differentiate between a 'contract' and a 'written statement of employment particulars'. The latter is NOT a contract but it IS a legal entitlement. It consists of two parts. The first, called 'the principal statement', must be given to the employee on the first day of his/her employment (or, of course, prior to that). That has to be supplemented by a wider written statement within two months.
I'll save myself a lot of typing (or copying and pasting) by letting this link explain what all that means:
https://www.gov.uk/employment-contracts-and-conditions/written-statement-of-employment-particulars
So the employer is correct in that you can commence your employment without a contract, per se, but Mr S is correct in stating that you must be given a written statement of the key elements of your employment.
BTW: It's not a 'mad hour'! The night is yet young ;-)