Quite a bit of this information isn't entirely correct. You may not have to stay until the end of your contract, but you would have to stay for much of this period if you want your redundancy pay.
There is something called counter-notice. There is a very good explanation of it here: https://www.usdaw.org.uk/Help-Advice/Workers-Rights/Finding-Another-Job/Negotiating-Early-Leaving-Date
The important thing to note is that this only applies if (a) you have definitely been served notice of redundancy and (b) you are within the statutory notice period (in your case 12 weeks back from the confirmed date of redundancy). If your redundancy package is more than the statutory, you should also check the policies - in some cases policy may dictate that only statutory redundancy will be paid if you leave earlier, even with agreement.
The employer can still refuse to let you go earlier, but you have some legal leverage - most employers won't take that on as it is just more cost and hassle for them.
You should also check very carefully any policies about existing job offers - technically, if you have a job offer in hand before the statutory notice period starts, then you aren't entitled to redundancy pay even if you have been served notice of redundancy. That is true even if the new employer agrees to wait until your notice expires. Keeping offers a secret is pretty hard given take up of references etc. This is because in law redundancy pay is not a "reward" for service or for losing your job. It is compensation for loss of income to enable you a period of time to replace that income. If there is no loss of income, it is deemed that compensation is not due.
It is worth spending some time getting clued up on the deatil around redundancy. As someone else said, there are rules around alternative employment offers that can come into play, and there are a lot of myths around what suitable means in law - it can include jobs that are lower paid or on worse terms in some circumstances.
Finally, whilst redundancy pay may look nice, do remember that if you haven't got a job it will run out very quickly, and the economy is volatile at the moment. If you need to work, jumping before everyone else is also looking for a job may be wise. Only you can decide that. But you cannot assume that those jobs you are looking at now will be available when you want them, and long term gains may be better than short term considerations.