Other people have suggested ways for you to decrease outgoings, but I think you need to increase income as well (and you can, by a good anount).
Firstly DO NOT apply for maternity allowance as suggested by others, as based on what you said it seems to me you are entitled to statuory maternity pay. Assuming you satisfy the criteria for statutory maternity pay (essentially that you worked at the company before you were pregnant and provided your MATB1 form), your former employer needs to pay you this, it is illegal for them not to do so surely. You were made redundant at 35 weeks, which is well past the qualifying period (15 weeks before due date). Unless you have already been paid this, as a lump sum, as part of your redundancy package? But it would have to be the entire amount of SMP, taxed as such with NI deductions, and in addition to your redundancy pay. If that is the case, then you would be expected to use that lump sum as your income for the 9 months that would have been your maternity leave. Therefore you'd need to factor that into your monthly income, not just your partner's salary.
If however the company has not paid this to you (which appears to be the case), and you are entitled to it, they must still pay this in full. You as a (former) employee and them as a (former) employer cannot opt out of it even if they offered a generous redundancy package without the SMP, it is statutory. HMRC can pay this to you directly and then recoup this from the employer if they fail to pay it. You need to get in touch with your former employer and ACAS if you haven't received it.
Furthermore, make sure your child benefit is sorted, as every little helps. When mine was tiny it covered all the monthly baby costs for a few months, as I believe it's £84.60 every 4 weeks.
And based on your partner's income I don't see how you couldn't be entitled to universal credit. You must be. Please apply ASAP. My salary is higher than your partner's plus my partner works part time, and like you we have one child, and we still receive universal credit. We are not in an expensive area either. The taper rate and work allowance has very recently changed so you don't lose as much universal credit when earning, which also helps. You will need to open a UC claim as a couple so it takes into account both your earnings.
Check your energy supplier and if you are eligible for the "warm home discount scheme", it gives you £140 credit off your electricity bill over the winter (by 31st March). You may have to get yourself on universal credit first depending on your energy company's scheme's requirements.
Once on universal credit, you should be able to apply for a Sure Start Maternity Grant, you can apply for this until your baby is 6 months old so better to apply sooner (so sort out your UC!) The scheme gives you £500 cash in your bank account to help with the costs of being a first time parent on a low income.