IANAL - So take the following with a Grain of Salt.
My understanding of UK employment law is that it is very well developed and mature - even without a contract, you have a number of legal protections.
If this is a Minimum wage job, I'd be less concerned about a lack of contract - since there is a statutory requirement to be paid at least the minimum wage. You are still entitled to Public Holidays, annual leave, sick leave etc. etc.
However, if I was promised a salary much higher than Minimum Wage - and I didn't have a contract to prove it (or some other documentation in writing) - I would be more concerned. If you have an offer letter or email or txt message outlining what you will be paid, that should suffice in lieu of a Contract (if push came to shove).
Additionally - a few days isn't ringing alarm bells for me. It's not unheard of to get a new Contract and take 1-2 days to read through it before signing. Admittedly normally this is done prior to starting somewhere - but that's not always the case, depending on the role and circumstances.
The final thing is, so long as they pay what was agreed, I'd be less concerned per se - If they honor that part, I'd be more comfortable waiting for a Contract.