There are several dog vaccines licensed for use in the UK and these have slightly different schedules for when each part is given. Your Dog vet Vicky Payne explains the protocol that her practice uses here, but it may not be exactly the same as the one your local vet uses.
We usually start puppy vaccines at eight weeks when the puppy is in their new home. The first vaccination covers leptospirosis (I use a four-strain vaccine, L4, as we are in a high-risk area, but in some areas a two-strain vaccine, L2, might be sufficient) and distemper, hepatitis, and parvovirus (DHP).
After two weeks, and when the puppy is at least 10 weeks old, we give the second DHP vaccine and one for kennel cough. Because it is so important that puppies get to experience the world at an early age, we advise owners they can take their puppies to training and for short walks a week after their second vaccination but to avoid water, parks, and other high leptospirosis risk areas (places with rats).
Two weeks later (four weeks after the first injection) the puppy has their second leptospirosis vaccination and can go anywhere after another week. If puppies are over 10 weeks when they first come in for vaccination they only need one vaccine for DHP and two against leptospirosis four weeks apart (two weeks for some brands).
If there is a high local risk of parvovirus, or in breeds where antibodies from Mum last longer and can stop the vaccination being effective, an extra DHP vaccination might be suggested at 16 weeks.
If your puppy has had a vaccination with their breeder make sure you have a copy of the record when booking with your vet so the vet knows what other injections your puppy needs and when.
The best advice I can give is to contact your local practice who can give you information on the schedule they recommend, as well as local risks, to help you balance protection against disease with training and socialisation opportunities.