Hi OP, I hear you, it's a hard slog and about to get worse. I am an anaesthetist - now consultant w 2 DC, but had DC1 with 3 years of training to go.
First of all, part-time is not your friend in training, as you already realised. At least if you go fulltime, you'll be able to afford a nanny and finish faster. Also, full time nanny is automatically part of your bubble (in a way that a part time nanny can't be as she has to work for more than one family) - something I had no idea I would appreciate last year as, like you, we also previously relied on grandparents (and still do, but less due to the restrictions).
Secondly, how much do you want to do surgery per se? I remember facing that choice many years ago and a wise senior trainee said to me that you want to be sure you like surgery 100%, not even 99% is enough. I realised that what she was saying, in my case, would have meant no children, possibly not even a partner unless they were prepared to drag around the country with me from post to post for years (I'm not in the UK). Is there any other path you'd like to try in medicine? Or para-medical - e.g. medicolegal/advisory/academic medicine/similar? You haven't done your final exams yet, if you retrain now, the beginning of the new training time is often gentler, so you'll find you have time for your children, then, as they grow up you can ramp up as well. There is no right answer here, you need a few hours to yourself (hard right now as well!) to think this through. Is there a mentor you trust and can speak to about this? Preferably more than one, as sometimes senior medics can be a bit tunnel visioned too - you need to see more than one tunnel then!
Thirdly, you need to have a chat w DH - he needs to be more supportive if you want to stay in medicine or any other career. There are ways for him, but HE needs to look for them/negotiate for them/basically put his money where his mouth is in terms of allowing both your careers to thrive.
There is no way I could be where I am without my DH, but he's tenure track academic whose hours are more flexible than mine - he does all the drop offs and pick ups and reading etc. He does work in the evenings when the kids are in bed. He's no saint and I find I still have most of the mental load, but he does try his best most of the time.
Most importantly, we have a full time nanny for DS2 (and DS1 after school). We pay her extra to do cleaning/tidying/hanging up & then folding away laundry (DS2 still naps solidly + she's training him to be a helper when he's awake, b/c even sorting pegs is fun at his age). This is something I specifically screened for when interviewing - she needs to be "your" person.
Lastly, a word of caution for the future, part-time doesn't always work for consultants either, depending on your area of subspecialty. My employer technically can't refuse if I try, but it would make my actual working conditions (and pay) very unattractive. After all the hard work and sacrifice I'm not about to give that up.
Basically my advice is absolutely don't quit a career, be it surgery or something else, but you have some introspection and serious conversations ahead of you.