You owe me my notice!

I don't think your unfair dismissal prospects are great. If you have a year's service, then arguably you could bring a claim - but the maximum value would be four weeks' money, and that would be reduced by the fact that you had started work elsewhere almost immediately.

Breach of contract/unlawful deduction would be a better bet. However, they may argue that you 'agreed' to a consensual variation - i.e. an immediate termination of employment without any notice expectation or expectation of payment in lieu. You will say you only did that on the basis that you would be paid in lieu - and it is likely to come down to the strength of oral evidence.

I think it is definitely worth pressing - could you get a local solicitor or the CAB to draft a letter before action, in the hope that a 'scary' letter will convince them to pay up? If that doesn't work, then lodging tribunal proceedings (which I'd do in the alternative for breach of contract and unlawful deduction from wages) is not difficult.

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