Just to clear a few things up:
There is no legal requirement for a written contract of employment, only for a written statement of particulars of employment, which the employer must provide within two months of the start of employment (not four weeks as stated).
An employee working without a (written) contract is not 'in theory working illegally' and certainly does have employee rights, since a contract of employment may be unwritten, although that employee will clearly have greater difficulty demonstrating all of the terms of the contract.
An employer does indeed have the right to receive notice from employees. Statutory minimum notice to be given by an employee is one week if he or she has been in continuous employment for at least one month. If employed for less than one month, the common law requirement of reasonable notice will apply.
The 'industrial tribunal' was replaced by the 'employment tribunal' some time last century.
Reassuring to see HR so on the ball.
Julia Hollywood is a freelance human resources consultant based in Essex.