>>> So when he goes to stay at his dad's in Essex, it will cost him?
Not if he connects his phone to his dad's wi-fi. Whatsapp itself is always free. It's only data carried over the mobile phone network that gets charged for. (So you can use Whatsapp without incurring any charges anywhere that there's free wi-fi, such as in McDonald's). However, if someone is using a 'bundle' then all data up to a prescribed limit is included anyway, so there won't be any additional charges unless that (usually very generous) data limit is exceeded.
If your son will be playing LOTS (and I really do mean LOTS!) of online video games and/or streaming LOTS of YouTube videos, your best option would be to seek out an 'unlimited data' deal. (You can get an unlimited data 'bundle' with Asda Mobile for £20 per month). Otherwise looking for a deal with a fairly generous data allowance (such as the 10 GB that I get per month for a tenner) should suffice.
At one time it was much, much cheaper to go for a contract rather than to use PAYG but these days, with many phone service providers offering really good value 'bundles', it can actually be a lot cheaper to use PAYG.