Who will benefit from the rate cut?
This is a difficult question and, as is often the case for VAT queries, "it depends"! This point is in part a legal issue and in part a commercial issue so the first point is to check the contractual position.
In the VAT legislation, there is a provision which states that, where a contract is made before a rate change but the contracted supply is made after the rate change, that, unless the contract provides otherwise, the price is increased or decreased by an amount equal to the change. So, for example, if the service price was agreed on 1 July but the tax point (i.e. earlier of the service taking place, invoice and payment date) is on 20 July, this provision applies and the reduction must be passed onto the customer.
However, where the service is booked after 15 July, if, contractually, you agree to sell something for £100 plus VAT, you must pass the reduction onto the customer. I am not a lawyer, but would expect that where the contract says "£100 plus 20% VAT", you would also be obliged to pass the rate change on. In these cases, only £105 can be charged.
Where there has been no mention of VAT contractually, or anywhere else within the agreement or price offer (or indeed the price is quoted as "VAT inclusive" only with no mention of the rate), then there is no obligation to pass on the rate change. This may then become a commercial decision and of course there are arguments both ways.