If you have been continuously employed at your place of work for less than 2 years then your employment rights will unfortunately be somewhat limited. Most importantly, you will not be protected against unfair dismissal. This means that your employer can dismiss you for more or less any reason, and without following a fair procedure, as long as their decision is not based on discriminatory grounds (i.e. because of gender, race, religion, age, a disability, sexual orientation, etc.) or because you were trying to assert any of your statutory rights (e.g. requesting maternity/paternity leave, etc.).
If the dismissal had nothing to do with any of the above exceptions then you would not be able to challenge it and your only protection would be if you were not paid your contractual notice period, because unless you were dismissed for gross misconduct, you would be entitled to receive your contractual notice period. If you did not have a written contract in place you would be entitled to the minimum statutory notice period of 1 week. Your employer would either have to allow you to work that notice period and pay you as normal, or they will have to pay you in lieu of notice.
If you were not paid your notice period when you were due one, that would amount to wrongful dismissal (which is different to unfair dismissal) and you could make a claim in an employment tribunal to recover the pay for the notice period that you should have been given. There is a 3-month time limit from the date of dismissal to submit the claim.
This is your basic legal position. I have more detailed advice for you in terms of the steps you need to take if you are to pursue them for this notice period, which I wish to discuss so please take a second to leave a positive rating for the service so far (by selecting 3, 4 or 5 stars) and I can continue with that and answer any further questions you may have. Don’t worry, there is no extra cost and leaving a rating will not close the question and we can continue this discussion. Thank you