OP, some issues to think about.
You haven't said if your landlord will allow pets. Any reputable rescue will insist on you bringing written evidence that your landlord would allow a pet.
Can you afford a dog walker to come in during the middle of the day - a professional, not just a friend or relative?
You mention that your remaining cat still comes to eat - can you help to rebuild this relationship?
I would actually counsel against you getting a dog at this stage, with your working patterns. As others have said, dogs can be combined with a flat, and a cat, and a young child, but as the sole adult, with your working patterns, it's asking a lot. What would happen for instance if you had to overtime at work or were asked to do some extra hours?
There are many, many less demanding pets to think about - how about a rabbit, a pet rat or something similar?
You can still get a doggy fix - how about volunteering for a local rescue? Most are delighted to welcome volunteers and organise lots of nice social events that are suitable for families and children, things like summer picnics, BBQs, fun dog shows, etc.