If the AST allows for rent increases (i.e. states how this can happen), then follow that. If not, you don't need to have a new contract, you can serve a section 13 notice which sets out past and future rent parameters in a straightforward way. Template on UK Gov website.

Regarding the deposit, you're only talking 2 weeks extra. What makes you think this might now be needed (given this is someone you know) versus the upheaval this will cause regarding signing a new contract and protecting the extra deposit, what the tenant might read into your intentions by increasing the deposit and how much those two extra weeks will actually be worth to you in the grand scheme of things if your arrangement does actually go South?

You can do what you suggest re increasing the rent; no notice needed; just date the new agreement to run in from the old one. But no need; you can just let the existing agreement go ‘periodic’ (month-to-month), at the new rate.

The amount of the deposit is specified in the tenancy agreement so if you wanted to increase it, I suppose a new agreement would be the way forward. It’s unusual and weird though. I don’t know what the protocol is for protecting the extra, but yes, probably safest to do what you suggest (and serve new prescribed information).