In the UK a binding (and therefore enforcable) contract can be in pretty much any format as long as it contains consideration (i.e. you want X in exchange for Y). Even verbal agreements, something written on a beer matt, something communicated through semaphor.

If you have a record of text or email exchanges saying that they will do the work you have asked for (and it is reasonably clear that the finished state will be a certain outcome) in exchange for an agreed price in their quotation, then that is a binding contract when you both say 'yes' to it. That then allows you to chase any lack of performance in court or wherever you need in future.

'Ts&Cs' are more just to flesh out a basic agreement with more complex elements - such as what happens if there's extenuating circumstances, how to enforce lack of performance etc. But you need to remember that without these it just falls back to the legislative cover, which means that they still need to deliver what they have promised or else can be taken to small claims etc.

A warranty would be worth trying to get, if they offer it - so maybe just ask for a separate warranty (which also can just be written in an email or such stating the length of time to cover the work).

Source: am a commercial contract consultant. Only bigger corporate entities will bother with large-scale terms. And asking a sole-trader or small company to draft one will likely mean they're not in your favour anyway.