If you do not have an agreement in place and just start to pay less than full mortgage payments, this may certainly impact your credit rating, which while it may not seem important now will be an issue if you are later in a position to take over the mortgage.
If you and your ex can agree to approach lender together, see if you can get a moratorium on payments and/or arrange an interest only deal for the short term. You may also be able to move to a lower interest rate if your Loan to Value has decreased. This may give you some breathing space and even a small amount of money could be the difference in being able to breaking even each month. If both names are on the mortgage, it's highly unlikely that a lender will make modifications without both parties signing off on it.
If you can obtain even part time work (around 20 hours per week) you would be entitled to Working Family payment (previously called FIS) - this will top up your salary and is not included for medical card . (Also not taken into account for mortgage application, but hold still on that )
There is a National Childcare Scheme which you should absolutely look into. I think there are a lot of hours for younger children and even school going children (older children being means tested). Get this setup - try get some hours to give you a break and time to sort stuff out and also to have childcare setup in advance of looking for a job. NCS are very slow so give time to find a provider who accepts this and understand what hours you can avail of. Might be worth giving them a call.
For now, try keep the wolf from the door but remember that this is the lowest things are going to be. You will get back on your feet and prepare yourself for that. Look into the childcare to give yourself a chance to tackle the other things. If need be go the rent a room route, but maybe try get someone who won't be there every w/e so that you can maintain some privacy.
best of luck