Running any business from home runs the risk of contravening planning regulations. In general, local authorities neither know nor care about people who just use their home as a base (e.g. plumbers) or who actually work at home doing jobs which don't annoy (or have the potential to annoy) their neighbours (e.g. typists).
Advertising a dog-sitting service (where you have the dogs as guests in your home) might come to the attention of the planning authorities. (e.g. a neighbour might be concerned about the possibility of excessive barking). It's likely that they'd rule that your activities amounted to unacceptable business use of residential premises.
Dog walking is far simpler. If you collect the dogs from their owners homes, there would be no reason for your neighbours or the planning authorities to object.
Basically, there are almost no 'rules & regs' relating to dog walking. You can simply advertise your services and get started. However, it would be wise to have insurance and you should also be aware that some councils now have bye-laws which state the maximum number of dogs that any one person can be in charge of (e.g. in a public park) at any one time. See here for an example:
http://www.wandsworth.gov.uk/Home/Environmenta ndTransport/Dogs/dogwalking.htm
Your local council's recreation department (or the dog warden service) should be able to tell you if there are any local rules you should know about.
This might give you some inspiration:
http://news.bbc.co.uk/1/hi/business/1788581.st m
Your vet might be able to offer advice and, once you've convinced him/her that you're properly insured, you might be allowed to advertise at the surgery. (Don