Not in Rome but in northern Italy although I used to go there often for work. I can give you pointer re cost of living in general and also maternity in Italy as I gave birth here.
Childcare for a 2 yo will be in a nido, childminders are rare in Italy and tend to be in the north. However, if your child is nearing 3 they would go to a scuola materna, which covers 3-6. There are public ones, often attached to schools, which are free and you only pay fir the lunch. Otherwise there are private options, from the state-subsidised catholic schools, which cost around €200 a month, to non religious community schools and pricier private options, such as bilingual schools. There won’t be any formal learning, it’s learning though play and many state pre schools use Montessori methods. Primary school proper starts at 6.
Maternity care in Italy is excellent, more tests and scans than what is available on the NHS and all free. Mortality rates are much better than the U.K., one of the lowest in the world. It is more medicalised, though, and midwife only care is rare. I had monthly appointments with my gynae, monthly blood tests for pretty much everything, plus 3 scans (if you go private you get a scan every month) and hospital monitoring when you near your due date. Maternity leave is also very good here if you work.
Cost of living in Italy is cheaper for some things and pricier for others. Cheaper: accommodation (although Rome is one of the pricier cities), public transport, eating out and entertainment. More expensive is food shopping, but it does depend on what you eat and the food does tend to be better quality and less processed, more in season stuff. There are loads of local markets, bakeries etc. which don’t cost a lot, plus decent wine is also cheaper. Car owning is also pricey, but in Rome there loads of car sharing schemes available so you don’t really need one.
Rome is a fabulous place, quite apart from the city itself and the amazing history you’re near the beach and surrounded by hills. The Apennines are not far away for trekking, plus you’re within easy reach of Tuscany, Umbria, Marche, Campania, Puglia. Trains are cheap by U.K. standards, including the high speed network, which is very efficient.