In the UK, no specific law states how many dogs you can walk together at once, either by a professional dog walker or an owner.
However, although all councils agree that the dogs should be ‘under control’, it is worth checking with your local council, who may have particular byelaws. NarpsUK (The National Association for Pet Sitters & Dog Walkers) has terms and conditions for its dog-walking members, including that ‘no more than four dogs should be walked at once’. Four dogs per handler appears to be a reasonable amount. However, it can depend on each dog’s temperament, size, strength, age, and health.
The critical factor when walking more than one dog is that each dog feels comfortable close to the others, and the dog walker has control at all times. We should all pick up after our dogs and this task obviously increases with a number of dogs, so a responsible dog walker should observe each dog closely.