I'm sorry to hear of all your troubles.
TBH I don't think it's a good idea to try and do a job interview with your baby roaming around, you'll inevitably seem distracted and unprofessional - I know you did try and reschedule but when they said that was not possible I'd either have tried to find someone to take her just for the time of the interview, or gracefully declined and asked them to bear me in mind for their next round of recruitment. I don't think you didn't get the job because you are a mum per se, it's more that you weren't able to give the job interview the focused attention they wanted at short notice. I think they likely would have felt the same if their candidate was dividing their attention for whatever reason, say it was someone who had a large and lively dog barking in the background, or who for whatever reason needed to do the job interview in a noisy cafe and kept being interrupted by people walking past - clearly the company wanted someone able to drop everything in their life for this short-notice interview and weren't prepared to flex that, so it wasn't personal to you. Now, is that the kind of company you want to work for as mum of a young baby, probably not. Would a kind, understanding and supportive company have that expectation, no. Does it give you an insight into what they'd be like as an employer - I'd say yes, you've probably had a lucky escape there.
I feel for you, applying for jobs is such a pain and it's really annoying how companies expect you to be able to prioritise them over absolutely everything else (including your existing job if you have one) for very short notice interviews, but they don't or won't commit to you in return. But on the plus side not all companies are like that, plenty have good flexibility and support for people with all sorts of different backgrounds and needs, so I think you just chalk this up to experience and move on - absolutely don't throw your hands up and say 'oh well I can't be a mum and work' because that's totally not true. I do think though that as an investment in your future working life, seeing what you can do to establish some form of 'emergency' childcare, whether that's a local teenager that would come and do an hour or two's babysitting at short notice for cash, or an emergency nanny from childcare.co.uk or similar, or a local friend or mum who would do childcare 'swaps' or even a gym creche or flexible childminder or similar would really pay off, as I know from experience that nursery usually won't have them for sessions other than their fixed regular slots, but life isn't always like that, babies get sick and can't go to their normal childcare or something happens at work and you need to give it some attention or you get trapped due to traffic meltdown or whatever and it is really really good to have some flexibility to avoid the stress on you as much as anything else. The baby's father (if around) of course also ought to be equally committed to looking after her while you both work.
Good luck!