That sounds less like Daycare and more like live-in Nanny, and it is a big ask. You are effectively working a job for them.
It'll depend on how comfy you are at their place, how happy the household is when they are back at home (low stress?), how "easy" the child (children?) is/are.
Honestly, one other question they need to ask is are they good with you being there. I assume they don't live in a huge house. The convenience of being able to work will be very helpful for them, but their marital privacy is going to be shot. Not saying you aren't good company, just that certain aspects are going to be different and could cause stress.
If you are not going back and forth daily, I think you can work around the weather related things. But this whole arrangement will require a good deal of sacrifice all around. May work out swimmingly though.
I'd drive up and spend their work week which they think they can arrange for 4 days/week for at least one if not both of them.
They live in a townhouse that is sufficient for the three of them (one small baby plus two parents). Three bedrooms, two baths although the second bath is in the basement and is .....not great but works. Guest room is small but sufficient. I don't need a lot.
I am most concerned about the hit to privacy, especially for them. Because the baby is so young, I am less concerned that I will 'do things wrong' according to the most up to date standards of infant care/their preferences. I'm happy enough to do things their way. I totally get that it's their home and their child. When the baby is older and more mobile and exerts her own will over her environment, there might be some conflicts, potentially. The last time I was up for a couple of days they 'complained' (not really) that I had spoiled the baby by holding her all the time. She was 6 weeks old so no, it is not possible to spoil the baby by holding her. It wasn't really a complaint--they are happy that I am so happy to just hold her, change her, feed her, etc. and hand her over when they want her back.
I'm also a bit concerned unless there are clear expectations for exactly what I will be doing. Do I go grocery shopping? Prepare dinner or other meals? Laundry? Vacuuming? I can clearly do all of those things, albeit less efficiently if I am also caring for a young infant.
Also: what's the back up plan when roads between my house and theirs (120 miles one way) are terrible? I drove back and forth to work no matter what the weather for 15 years before I retired, about 100 miles/day. Towards the end, I confess that I felt it was less and less worth it. But at that point, I was working for money. If I do childcare, it would be a)for no money and b) for my grandchild. I'd happily part with a kidney for that baby. I'm just concerned about a back up if weather were awful. I figure I could just leave some stuff in the guest room to make it easier/lighter to travel back and forth. It isn't luxury accommodations but it's fine. I'm easily entertained and have no problem retreating to curl up with a book. I think they are worried I'll be bored. I'm worried about their privacy.
The relationship is not that old: baby was a surprise. I think we just learned of the mother's existence about a year ago---and by we, I mean the entire family and friends network. I've already mentioned to them that if/when they are ready, we'd take the baby for a weekend or so so that they could get away or even do a staycation kind of thing. Something they had hoped to do before baby arrived but weren't able to manage it. It was a high risk pregnancy so they were reluctant to travel away from her doctors. Very understandably.
My husband and I were in a somewhat similar position when we first got together, with our own surprise pregnancy---not a secret to any of the now adult kids. They know we had nothing, struggled for years financially---and have been pretty happy most of the time. I won't lie and say that the over 40 years together have all been rainbows and bluebirds. They know that, as well. The advantage/disadvantage that we had is that we were young and pretty stupid and that we lived more than 500 miles from our families, so we had to figure things out for ourselves and there was no one around to either help or criticize. They are in their 30's so they are not as young and stupid as we were nor are they as poor. We're closer than any of our family ever was, but hers is quite far away so we're it for local family. She's more outgoing and has a wide circle of friends which is a good thing/bad thing for their relationship. One of her friends was living with them until about a month before the baby was born so they really have not had much time alone to figure out how to be a couple. That said, my observation is that they really do work well together. My son has even mentioned that he realizes that they work together on projects together better than my husband and I do now, after all of these years.
It's a bit premature. I'm heading up for a few days and it may be that there are more talks.
Bottom line: they know I will step in if/when needed, and will fill in gaps in care if needed.
I'd be less like nanny and more....Granny?????
There are other things being discussed but they are longer range. Hubby has been dead set against ever moving but last night mentioned that it is possible that we could move closer to them. Also she'd like to live in our community because it's quieter and there's more nature/less day to day noise and stress around. Of course that would mean both of them changing jobs, selling their home and purchasing another home, with the potential to temporarily live with us if necessary. That's a LOT with a brand new baby. And I'm writing this as someone who has relocated across state lines twice with young kids, twice with a year old baby to places where we knew no one and had no friends/family. Also during the days before there was an internet so it was not fun looking for houses to rent or buy.....but we did it. I think we just never considered not succeeding. Stupid? Lucky? Probably both.