I'm a solicitor with around 8 years of experience and have a young child. There are a lot of solicitors who would try to put you off a change to law. I think it can work ok with a family but there are a lot of caveats to that.
It is common to work 4 days at my firm but difficult to do less than that. The 4 days are quite long days and you sometimes have to work a bit on your days off.
The real difficulty for you is likely to be getting a training contract and then getting through it. It is very competitive, particularly at the moment as some training contracts have been delayed due to covid. Law is more accepting of career changers than it used to be but there is an expectation, at my firm at least, that they bring something to the table from their previous experience so in some ways the bar is higher.
I haven't come across a trainee working part time before and you have very little control over your workload while you are training - if you're in a transactional seat at a medium / large firm you might be asked to work late into the night with no notice, for example. Most trainees are able to do this as they are young and don't have family commitments.
Whether law is sustainable with a family long term very much depends on the area you work in and your firm. Generally, law is starting to wake up to the fact that a lot of lawyers are women who will leave the profession if they can't work flexibly but there's a way to go and this tends to apply more to experienced solicitors than trainees / newly qualifieds. I wouldn't personally try to work in litigation or transactional work at a large firm with a young family but plenty of people do (usually using nannies for childcare).
That said, I find my work interesting most of the time, I work hard but not crazy hours, I have plenty of family time and I am paid very well with the potential to be paid extremely well in future if I go for partnership. It has taken a long time to get there though, I have chosen my speciality carefully and there were times earlier in my career where I was really stressed by work and working very long and unpredictable hours.
If you want to work at a larger firm it would be better for you to try to get the training contract before doing the conversion course as they would probably fund it for you plus you'd be guaranteed a job afterwards. The conversion course isn't much use on its own if you don't go on to qualify as a solicitor.