In a true emergency the scenario I'd see playing out is:
Child gets really sick
Childcare calls ambulance
(If you're not really close to the hospital from work) Ask/yell at a coworker to drive you to the hospital
Otherwise, childcares just tend to keep sick kids in the office until parents get there.
I plan on using the bike trailer when my son starts preschool (4yo) next year, my baby girl will be almost 1 when he starts, and the bike trailer is still a bit big for her (not supposed to use under 18mo) but she'll be fine for the 2km each way ride. I just put them both in it for the first time 2 days ago (baby is 9.5mo now) and took them for a few laps around the suburb and they loved it. I'm sure they'll fight as they get older, or maybe there will be so much to see they won't!