I would agree that most employed jobs where working from home is offered would expect you to have childcare in place. Once they start moving and talking it will be very difficult to do anything with a toddler present.
An obvious point...but have you done a spreadsheet of all your essential outgoings and then income from DP (assuming you have one) and child benefit (if you get it) and then look at different incomes/childcare set ups to see what works. What do you need to earn? I mention this shouldn't you just be looking at your income and childcare, it should really be the family income and expenditure.
Working from home part-time could enable you to use a childminder rather than a full day at nursery so that could save you money that way.
Timewise jobs (part of the Women Like Us social enterprise) offers pt roles. There are often jobs on the MN jobs board. And all jobs board offer pt roles, I like indeed.co.uk as it searches most boards for you.
If you want to earn money with no childcare then a small pt job might work if you can be flexible when you work eg during sleep time/at night. The NCT pay their antenatal course admins about £5k a year - I think - (or they did when I was involved in volunteering) to set all the courses up, deal with bookings etc. Not a fortune but if you can do without childcare etc. Or maybe some of the selling franchises Avon etc but I do question how much you make.
You will be surprised what sort of small jobs are out there when you look. When I contemplated trying to work without childcare I saw a paid evening moderator position for a cancer support board and a 5.30am - 8.30am receptionist job for a local builder.
Could you freelance in your current field? That would allow you to work around the best value childcare set up. I now freelance. When thinking about earnings think in terms of 1/3 of your time delivering paid for work and then rest is admin, business development, marketing etc. So your fees need to take that into account.
I have lots of empathy, our childcare bill for 2 (4 days a week) is £1300 a month.
Good luck.