On the surface, it’s hard not to agree with every other poster - hot cups are dangerous to toddlers, 10 cups in 10 hours is taking the piss, but this is REALLY a situation where it’s hard to judge without seeing what’s really happening because some of the things you’re saying, OP, make me think that yes, you are part of the problem? Let’s dissect.
Let’s start with the drinks themselves, and whether your nanny is a careless slattern (joking a bit, as I imagine this is all stressful).
More than once I've had to pull DS away from the table they are at and ask her to move the mug.
I understand that it’s very important to keep drinks and toddlers separate, but you’re saying that you’re stopping him before he reaches the table that contains the cup. Perhaps it’s exactly like this: DS’ hand is always centimeters from the table, grabbing toward it, when you stop him, and the drink is always centimeters from that; you’re giving her a chance but she does nothing. But I’ve also seen a lot of mums have their children several feet away from a table and the minute the child started toddling towards the table (still several feet away), the mum asks someone to move their hot cup from the table in a hurried, panicked voice. The ONLY reason I’m wondering this is because you later say:
I do try to step away and I've had my mum over as well to get her started whilst I've gone out to try to give her a chance with DS (which apparently worked quite well). But the disregard for safety is making it difficult for me to feel comfortable leaving them alone with her - I do struggle with letting go and do recognise that this probably makes it harder for everyone involved.
So when she’s on her own, left to do the job you hired her for, she was fine? No hot drinks left in reach, no taking the piss with so many hot drinks and ignoring the children? You say it worked quite well without you, despite the fact that your mother was still there (why was this, by the by?), and you struggle with letting go. So if, with the exception of that one time, which you know worked just fine, you’re staying around and not letting the children settle with her then... maybe she doesn’t know what to do, except sit around drinking tea, and she figures she’ll just keep her mouth shut until you start working and then things will be fine?
Because I think this is a bit of the issue:
I'm sure it's not been the easiest start for her as DS has been quite difficult - having spent half his life in lockdown with me and DH, he's not adjusting very well to having someone else in the house. And DD is still EBF and also has spent her whole life as a velcro baby with me, so there's not a lot she can do there either.
Er... what? If course there’s a lot she can do. She’s a nanny. She can take care of your children. But yes, you may have to hand them to the nanny and walk away. That’s what she was employed for and probably what she expected. She came highly recommended, she probably knows how to help children and settle them, but not while they can stay stuck to you if they choose. I imagine she was expecting to have sole care of the baby for 10.5 hours and, as I said, is finding it all exceedingly awkward.
Also, I have to admit that I found this a bit 
:
He is pretty unsettled (new nanny but also other stuff happening as well) so is understandably a bit clingy, but I do feel that if she got down on the floor and started playing he would join her.
So you want her to just start playing on the floor by herself, and then hopefully, he will remove himself from you and be more interested in her? Easier solution: remove yourself from the situation. If you truly feel unable to just let the nanny get in with childcare in your own house while you WFH, set up a nanny cam that you can watch while working in your office? There’s nothing wrong with that if it’s what it takes for you to feel comfortable.
I’m just really curious (stuck in bed, too much time to think about a stranger’s life 
). Because as you said, OP, she came highly recommended with good references, etc, and you seem like the kind of person who would be extremely careful about that so... if the issue is your inability to disconnect, you may have the same issue with the next nanny, or some other issue, and the next, until you sort out your feelings about nannies, childcare, boundaries, guilt about going back to work (if there’s any of that involved, snd there shouldn’t be! We all do what we think is right for our families), and whatever else is involved.
Good luck, @namechangedasouting. 
I’m sure this is incredibly difficult.