We do like going abroad on holidays and this is one reason I was reluctant to get a dog even though the DCs were begging constantly. But we have 3 sets of neighbours who we know very well and they could look after our dog whilst we were away so I feel that will be OK.
Have you checked that they would be happy to look after your dog? It's a big ask! It's always worth having a back up option - for me, DF does much of the dog sitting when I have to go on business trips, but I do use a local trainer / sitter to fill in the gaps.
Re training, I'm actually not sure what you mean besides toilet training and recall and 'sit' etc?
Socialisation is a big part of what puppies need - getting them out and about and having positive experiences from an early age, so that they can face the world with confidence. There's a surprisingly short window in which to get them socialised.
Good quality training classes are a good place to start; Dogs Trust run some nationwide (not just in their centres) that are open to all and have a good reputation www.dogstrustdogschool.org.uk/dog-school/
Again re grooming, I have no idea what's typical or average? I was thinking we'd have to wash him once a week and clean paws etc after walks. What other grooming is necessary?
Depends on the breed and individual. My short haired dog requires very minimal brushing, but is a mud magnet and at muddy times of year can require a water-only rinse in the bath after almost every walk. A shampoo bath every week would likely cause skin problems. Some dogs require professional clipping, some require nail clipping (mine gets enough exercise on tarmac to wear them down naturally), some need anal glands expressing...
I am open to looking at rescue shelters but am struggling to find one nearby that has puppies. We are in London. Battersea seem to have older dogs mostly and as we have no dog experience whatsoever I do not want a dog that has been treated badly as we are not experienced enough to deal with any issues it may have.
Don't discount an older rescue dog. Many have had great lives but a change in circumstances means the previous owner can no longer keep them - a change in working hours, loss of home, relationship breakdown etc. etc. Relatively few come in due to abuse / neglect. I don't know much about Battersea, but I do know that Dogs Trust have particularly strong behavioural support if you ever do run into any issues.
My neighbour recommended Many Tears dog rescue which is in Wales. They have a dog from there but I feel we won't know what we're getting in terms of potential health issues etc as often they do not know what some of the cross breeds are.
They're a foster based rescue, which means that the dogs are looked after in family homes by experienced dog owners, while they're waiting for new homes. That allows them to be assessed as to their character, have some foundation training put in place, and to be matched to a suitable new home. It's in their interests to find the right home for the right dog - both for the dog's sake, and because they don't want the dog to be returned unnecessarily.
That gives you a dog from a source where it has been assessed as an individual and matched to you from a responsible source. Your chances of a good match are far higher with a dog from Many Tears than a puppy, where you have very little idea what you're getting beyond the confines of the breed (or breeds, if you're looking at a cross).
Many Tears does have a good reputation and are certainly worth considering.
OK thanks, will have a look at the places you have suggested. The only slight issue is that I am being a little fussy about 'looks'. After falling in love with my friend's Yorkshire Terrier I really really want something very similar, small, cute, fluffy and cuddly.
Small, cute and fluffy dogs do come into rescue - but they tend to be snapped up quickly. Get registered with several rescues and if you settle on a specific breed(s), get in touch with the breed rescues too. As others have mentioned, puppies are often snapped up before they make it onto the websites. So long as the temperament is right and it's a good fit for your family, even if you were to get a dog that's mid-sized, mildly fugly, short haired and only a little bit cuddly... I'll guarantee you that within a few days you will have fallen in love with the dog and it will be your dog and the apple of your eye. It happened to me - mine is a cross that found his way to me rather than being chosen. Two breeds I never would have chosen, not exceptionally cute, not even slightly fluffy... but he's mine and I love him.