That is an Irish one
I don't think you'd be able to get an emergency one for this. Adult renewals are fairly quick if you can do it online. Child first time passport takes around 8 weeks
My adult renewal took less than 2 weeks so you should have plenty of time to get your daughter's. I think hers took about 9 weeks to come after I applied for it.
If you renew yours and then apply for first childs passport i think you have enough time till May.
New Uk passports can be issued in less than 2 weeks. My sister applied for Irish ones for her family and it took about 2 months and was very slow
You should be okay if you do yours online asap like tomorrow lol as the childrens one can take a while and this is the peak time of year that everyone starts doing the exact same as you, this time last year my babys one took 9 weeks
If you apply for an emergency passport It’ll be here within a week