If the contract contained no provision for terminating your employment by paying you in lieu of notice, then doing so would amount to a breach of contract. Unless you were dismissed for gross misconduct, which is not the case here, they would have had to give you the opportunity to work your notice period which would have taken you over the 2 years required for unfair dismissal protection. This means they should have then provided a fair reason for dismissal and also followed a fair procedure. At this stage you should contact them first to challenge this and remind them of your rights and their obligations. If they refuse to resolve this then you can consider the unfair dismissal claim, which must be submitted within 3 months of the date of termination.
A new feature in the employment tribunal’s claims process is mandatory early conciliation with ACAS. This requires prospective claimants to notify ACAS and provide details of their intended claim and they would then try to negotiate between the claimant and respondent to seek out of court settlement in order to avoid having to take the claim to the tribunal. It is possible for the parties to refuse to engage in these negotiations, or that they are unsuccessful, in which case they would get permission to proceed with making the claim in the tribunal.
If negotiations are initiated and settlement is reached, then the claimant would agree not to proceed with the claim in return for the agreed financial settlement.
The conciliation procedure and the form to fill in can be found here:
In terms of the time limits within which a claim must be presented, the early conciliation process places a ‘stop’ on that and the time between notifying ACAS and them issuing permission to proceed with the claim would not count for the purposes of these time limits.