There is no legal minimum age in the UK for leaving children at home alone (I'm assuming that this is while the parents are out at work?) but they are sailing pretty close to the wind doing that with a 10 year old and 13 year old.

https://www.gov.uk/law-on-leaving-your-child-home-alone

Especially because if children under 16 are left alone to babysit younger children then if anything happens it is always the parent who is legally liable - 'But I wasn't there' is an admission of negligence not an excuse or an alibi. In the event of an accident to either child, is the other one going to have the confidence and quick-thinking to know what to do? I've seen adults panic in the event of an accident and to be so paralysed that they can't even get their act together to dial 999. How can anybody put a 13 year old, let alone a 10 year old in that position?

https://www.askthe.police.uk/content/q455.htm

And besides - I remember when my younger brother and I were around that sort of age - we fought like cat and dog if there wasn't an adult around to mediate! If nothing else it seems like a good way to set siblings against each other!

I wouldn't. The 13yr old maybe, on their own.

I'd be too nervous. Don't use the oven, keep the door locked, no fire on.

No. If they're still at a very young age, I would never leave them on their own. Perhaps, by the time that they're at least 13 years old and above (ages of grasping their sense of responsibility), I may, but for now, it's a big no.