I am unaware of any employment type relationship that requires a written contract. There always IS a contract as you do the work and they pay you.
Actually agency/worker does legally require a contract and it's particulars are laid out in law as well - as well as all the stuff that came out with IR35 and carefully writing contracts so people are or aren't in it etc there is this from the conduct regs
Provision of information to work-seekers and hirers
21.—(1) Subject to paragraph (3), an agency or employment business shall ensure that at the same time as—
(b)it offers a work-seeker a position with a hirer—
(i)it gives to the work-seeker (whether orally or otherwise) all information it has been provided with about the matters referred to in paragraphs (a) to (e) and, where applicable, paragraph (f) of regulation 18; and
(ii)in the case of an employment business that has not agreed a rate of remuneration in accordance with regulation 15(d)(i), it informs the work-seeker (whether orally or otherwise) of the rate of remuneration it will pay him to work in that position.
(2)
Where any of the information referred to in paragraph (1) is not given to the work-seeker or hirer, as the case may be, in paper form or by electronic means at the time referred to in paragraph (1), the agency or employment business shall confirm such information in paper form or by electronic means to the work-seeker or hirer, as the case may be, as soon as possible and in any event no later than the end of the third business day following the day on which it was given to the work-seeker or hirer in accordance with paragraph (1)
The matters referred to in 18(a) to (e) are these
(a)the identity of the hirer and, if applicable, the nature of the hirer’s business;
(b)the date on which the hirer requires a work-seeker to commence work and the duration, or likely duration, of the work;
(c)the position which the hirer seeks to fill, including the type of work a work-seeker in that position would be required to do, the location at which and the hours during which he would be required to work and any risks to health or safety known to the hirer and what steps the hirer has taken to prevent or control such risks;
(d)the experience, training, qualifications and any authorisation which the hirer considers are necessary, or which are required by law, or by any professional body, for a work-seeker to possess in order to work in the position;
(e)any expenses payable by or to the work-seeker;
and
(f)in the case of an agency—
(i)the minimum rate of remuneration and any other benefits which the hirer would offer to a person in the position which it seeks to fill, and the intervals at which the person would be paid; and
(ii)where applicable, the length of notice which a work-seeker in such a position would be required to give, and entitled to receive, to terminate the employment with the hirer.