I don't start jobs without a written statement (signed contract)... but that's just me.
From ACAS (source)...
"The Employment Rights Act 1996 (amended under the Employment Act 2002) requires employers to provide employees, within two calendar months of starting work, with a written statement of the main terms and conditions of employment."
Without a written statement, a contract of employment does exist.
ACAS says the following: (source)
"All employees have a contract of employment which forms the basis of the employment relationship. In simple terms, an employee agrees to work for an employer in return for wages. A contract is made when the offer of employment is accepted. A number of rights and duties, enforceable through the courts, arise as soon as this happens.
Most employment contracts need not be in writing to be legally valid; a verbal agreement can be sufficient. However, writing down the terms of the contract can minimise later disagreements. The Employment Rights Act 1996 requires employers to provide most employees, within two calendar months of starting work, with a written statement of the main terms of the contract."
So your way forward is to rise with your employer the situation regarding having a written statement which describes your duties, hours of work, holiday entitlement, wage, what happens if you are sick, that sort of thing. If after gentle nagging... and several weeks have passed, alert them to the legal requirement.