OFGS...
Give your head a wobble.
Wild rats won't approach a human unless cornered, simply supervise your child outside.
Speak to both neighbours, arrange pest control TOGETHER, so that everyone is working toward the same goal the same way.
Tidy your garden, block off holes, (being aware you may trap rats IN as well as out), remove any shelter you can (logpiles, overgrown planting, rubbish).
Ensure your pest controller is aware that there are children and pets (between you and your neighbours) and they must NOT use the packets of bait designed for rats to pick up and carry to their nest, as they can drop these out in the open...
Ensure they use lockable bait stations placed out of the reach of children, and poison that contains Bitrex and blue dye as this makes it far less likely a child will consume any and bloody obvious if they do.
They should do the above anyway but I know a friends dog nearly died when the pest controller used packets and the rat dropped the packet in the open and the dog ate it.
For the anti-poisoners - sorry but this is realistically the only way to deal with an outdoor rat population.
The reason you are seeing rats in the daytime is that they have had a population explosion and they are struggling for food and willing to take greater risks to get it.
This may be natural, or it could be that one of your neighbours has a food source that has allowed the population to increase beyond the norm.
You are never more than 10ft from a rat, the fact you know this now doesn't change matters, you didn't die from it before you won't die of it now, just deal with it sensibly!