Long story short we have a load of "Cash Price" sale tickets in stock (small tickets with "Cash Price" written on top, and then you can write the price underneath), but we have been told in passing it's not legal to use "Cash Price" anymore, but I can't find anything online to confirm this.
This is an incomplete answer, but regardless of the state of statutory law in the U.K. and Ireland, most credit card providers, as part of their merchant agreements authorizing a merchant to accept credit card payments, prohibit merchants who accept credit cards from offering a lower price for cash payment than for a purchase using a credit card, despite the fact that in the case of a credit card payment, the merchant has to pay a processing fee to the credit card company that the merchant does not have to pay in a cash transaction.
There have been some lawsuits challenging the validity of this requirement, but to the best of my knowledge, none have been successful.
Generally speaking, however, it is not illegal to offer a different price if the merchant is paid all at once, as opposed to offering seller financing on installment terms (a different sense of the phase "cash price"), which is a different situation than when a merchant is distinguishing between a credit card payment of the entire price in one go, and a cash payment of the entire price in one go.
It isn't entirely clear from your question in what sense "cash price" is customarily used with those tags.
Similarly, it is probably permissible to offer one price for people who pay via either cash or a credit card on one hand, and a different price for people who pay via a check (which carries with it a risk that the check will not be honored), since that is not subject to a merchant agreement restriction, although my impression is that checks are used less often for payments in the U.K. (where they were invented) and Ireland, than in the United States.