tl;dr: You should repay the money as soon as possible.
You could still try to spin things as a misunderstanding, but that will become a harder sell with each passing day. The harms could go beyond 150 pounds-- perhaps you develop a reputation for dishonesty, making it harder to get future jobs. Or you end up with a criminal record, plus court costs.
If money is tight, maybe the former employer would agree to let you repay in installments or something. This, too, becomes less likely as you try to delay the process and/or keep the money.
I'm struggling to find any argument which suggests that you can or should be able to keep the extra week's pay. Let's review what's stated in the question:
- You only actually worked for three weeks.
Even if you were salaried (which isn't clear from the question), a week in which you specifically did no work and did not fulfill any of the responsibilities of the job (like showing up to work or following the employer's sick leave policies) is going to be hard to defend.
Even assuming that the notice period the employer requires is 0 days, you didn't give that notice. Whatever actions the employer might have taken in response (like instructing payroll not to pay you for some period) were foreclosed to them, because you told them nothing.
Since it seems like you already had that 4th week's pay before declining to return, it wouldn't be a stretch to suggest that you committed intentional fraud in not giving any notice so that you could keep that money. Whether or not that's actually true, it's the sort of thing the employer could claim, and I don't see that you have much of a defense or alternative explanation.
The question reads as though you didn't even bother to tell the former employer that you wouldn't be coming back, ever. Instead you simply ignored them until they fired you.
- It seems unlikely that you followed the employer's relevant policies
in any way.
I'll bet that the sick leave policy is not "just unilaterally don't show up, and vaguely suggest you were ill if someone asks you about it". Any protections you might try to claim for something which might justify what happened will be badly undercut by having not followed any of the relevant procedures. It's worse when there are multiple instances of it, and in this case there are at least two (sick leave and notice period).
- You did not sign the employment contract.
From my reading it seems that you think this releases you from responsibilities to the former employer. That may or may not be true (ask on Law.SE for better advice on that), but it also releases the employer from a lot of responsibilities to you. If they would have been required to honor your ad-hoc sick leave if you were an employee, but you never signed the contract securing that... well, the employer has some motivation to not let you slide here.
- You are retroactively explaining your behavior in ways that
contradict your reasoning at the time you left.
The whole "I technically was sick for that fourth week" line is problematic because you had already decided that the job wasn't for you and so (that's a direct quote) you did not return for the fourth week. This may be invisible to the former employer, however.
- You don't seem to have any theory which suggests you should get to
keep the extra money.
It's understandable that you would want to keep the money, but you haven't advanced even a single argument suggesting that you should be able to keep it. The entire framing of the question is more along the lines of "can I get away with it?", and not at all "I deserve to keep it." That's a problem in itself, but the former employer has some pretty strong arguments that you specifically should not get to keep the money (that case is described throughout the rest of this answer).