it's on the low end of acceptable.
You say that "There's no question that it will be used for games or internet access." and I hope that - father to father - you'll forgive me if I smile slightly at that. I have no idea how old your children are, but if they're the same as most others there will certainly come a time when you'll find a game being played, and Internet access is really an essential aid to homework at the secondary education stage.
For around £500 you would find a laptop that will have wireless networking built in, and it would also be games-friendly. If you have a broadband connection you could move to a wireless router later on, and the children would have the internet access that they will definitely need. Today's kids are growing up with the internet, it's going to be an integral part of their lives, and I see no reason to plan for anything else when buying hardware nowadays.
This computer will probably end up getting more use than you anticipate, and I believe it might be worth your while to consider hiking the budget a bit now, and saving yourself additional expenditure in the future.