I don't think any of the restaurants allow strollers. I could be wrong, though. I've just never noticed strollers in any of the restaurants.
The way the tables are positioned in many of the restaurants, it would be way too crowded w/ strollers. The servers would have an awful time trying to navigate!
I've been to Le Cellier, and there's no room for strollers!
Another thing to consider w/ the signature restaurants is that the meal typically takes longer - especially if you're having appetizers, entrees, & desserts.
They do have things to keep older children occupied - like the activity menus. However, I'm not sure that will keep very young children interested for long.
My 2 older children are absolutely fine in any restaurant & have been for years. We went to our 1st signature (California Grill) in 2008 when DS was 6 & DD was 8. Not a problem at all. Prior to that, we mainly stuck to character meals & the "family-type" restaurants like WCC & 'Ohana.
For our upcoming trip, we will also have a 2-year old. He's okay in restaurants, but I still worry about how he's going to do - especially, after being in the parks all day in a stroller & not being able to run around. He may be great in restaurants here at home where he's been playing & running around all day long, but I worry about how he's going to do in restaurants after sitting in a stroller, being held by DH or me, & sitting on rides all day. We were going to Hollywood Brown Derby for our Fantasmic Dinner Package, but, the more I read, the uneasier I got... the Brown Derby just seems like a more quiet meal w/ less of an appeal to children. So, now, I think we're switching to Narcoossee's - it seems more child-friendly & more casual. Plus, I love their Nemo cupcakes that have on the children's menus for desserts! And, if he gets ansty (I'm hoping he doesn't!), DH or I can take him out on the deck. However, I still haven't made up my mind. We may just do Kona Cafe instead.
I also always try to schedule our dinners for between 5:00 & 5:30 pm - more of a "family" meal time, imho. And save the later times for those dining w/o children & wanting a more quiet experience. Due to our schedule for the day, I think I'm going to have to make our Narcoossee's ADR for 6:00 pm, & that's really the latest I'll go.
Jiko does look a wonderful restaurant, & I can't wait to try it myself! But, from things I've read & seen, it just looks like 1 of the more quiet signatures, & so we'll wait until baby DS is a little older. That said though, I've also read reviews of Jiko where the kids were absolutely welcomed & had no problems at all!
So, I guess what I'm trying to say is you know your children best & to use your own good judgment when choosing restaurants.

I would say that well behaved children are welcomed in all of Disney's restaurants except for Victoria & Albert's & Bistro de Paris. It's just important to take all the factors into consideration when choosing restaurants & to not forget about the other diners!
