As per the first tool, you don't have to sign the new lease and it automatically converts to a periodic tenancy. There is no need to sign a periodic tenancy, it happens automatically (so long as the last agreement was signed after Feb 2021).

They can still put up the rent, but only by giving 60 days notice and only if they haven't done it in the last 12 months.

In any case, you might want to leave, in which case you need to give at least 28 days notice (and it can't end before the end of the fixed term). You might want to stay, in which case you can just not sign anything, it automatically converts to periodic tenancy, but 60 days after they give notice the rent can go up (the fixed term ending has no relevance on this - they still can't do it earlier than 60 days).

I don't think they can kick you out straight away if you let it lapse into a periodic tenancy? Hopefully you can find somewhere else within the 2 months before rent can be increased under periodic.