Only read OP's posts.
Switch is a good call for one console to share. There's a new "OLED Switch" with a bigger and better screen. If it'll be mostly played by itself in tabletop or handheld mode consider it, if it'll be mostly played plugged into a TV then consider the older one to save money.
"Tabletop mode": You just prop it up on the kickstand it has, take the controllers off the console, and play. So you can all sit round for multiplayer without needing to plug it into a TV. Really good for if you're travelling. Obviously a TV gives a much bigger display (and also the Switch runs in a higher graphics mode when docked).
I think other posts have covered the accessories. Second pair of Joycons for 4 player. Screen protector's a good idea; the Switch has a plastic screen which is smash-proof but scratches easily.
On games. They're not all the 40 or 50 quid price. On the eShop there's loads of cheaper ones. Eg I got Gear Club Unlimited, a decent enough racing game (not the best, but it's alright) for a few quid.
Online multiplayer requires a subscription. You don't need that for multiplayer on one console, or even on multiple consoles in the same room/house/picnic blanket.
Parental controls are kind of complicated. Make sure there's no way the kids could run up a huge bill - there are some Switch games with mobile-style money pit purchases.
That's the Switch. If you think your kids would prefer a device each, consider the 2DS/3DS range, though you'll be buying second-hand. It's a last-gen console but the games are still being sold.