Unfair dismissal claim unlikely as you have not been employed long enough - 2 year qualifying period for unfair dismissal cases not related to pregnancy or disability.
4 weeks and 1 calendar month are different time periods. So if a contract said that, then you would give the longer period of notice to be safe which is 1 calendar month.
However on the Friday of the 2nd week I suffered from a terrible migraine and didn't go into work.
That would be a sick day, unpaid unless contract says otherwise or your employer is nice. Many parents are lovely and do pay for the occasional day of illness even if contractually they do not have to do so.
Later that day I then got an email from them saying that my contract (that lasts a year) was not going to be renewed and that I only need to give one week notice.
That seems a bit odd. Statutory Notice is indeed one week but if the contract says a longer notice period then the period in the contract would apply.
If they want you to leave earlier, then they can give you gardening leave where you get to stay at home and still be paid.
You need to refer to exactly what is written in the contract. Then you need to set out you case clearly to them showing the timeline of events and your expectation of being paid.
If they want you to leave with immediate affect then they can give notice under the terms of the contract (though that notice period is likely to be the same as the one you gave them) or accept your resignation and pay you for that period even if they do not wish you to physically go to work.
If they fail to do that, then you could take it to a small claims court but the court would expect you to make effort in resolving it between yourselves.
Set out the timeline, set out your expectation to be paid and the date when you expect payment to be made. Request that a final payslip is produced which includes any outstanding holiday not taken (this may be a very low amount if you are now just in your second year) and request that a P45 is provided as soon as possible following production of that payslip. Provide them with details for ACAS so they can seek advice. They can also contact their payroll provider for details of the procedure involved for when an employee resigns.