Hello I am a solicitor with over 15 years experience. I will try to help you with this.
There is no reason why you cannot bring a claim against a vet for negligence for costs incurred and suffering experienced by you. Any award of compensation is unlikely to be very high. You could also complain to the RCVS http://www.rcvs.org.uk/complaints/ if your main concern is to have him investigated by his professional body, to ensure the future safety of other people's pets.
There are solicitors who specialise in this sort of work and a Google search will bring up a number of firms. I am not allowed to recommend any particular firm to you but if the firm also do clinical negligence claims and other professional negligence work then it is likely that they will know what they are doing.