As a long term nursery user and working parent I heartily sympathise. I had very limited Grandparent availability when my dc were ill and now it is non-existant as my parents are too old and I don't want to expose them to so many illnesses themselves.
I would however advise you to use this as a wake up call for the future.
Nursery attenders are exposed to lots of other children's germs and potentially will experience plenty of viruses, some of these such as chicken pox, hand foot and mouth etc may require more than a day or two off work for you. (Having got this sort of childhood stuff out of the way your child will however very possibly be rarely off school when they reach school age, having developed some immunity to bugs.)
You do need to have a plan however for chicken pox or whatever, keep a file of agency nanny details or whatever you may need. Have you asked nursery if there are any part-time members willing to come to your house on their days off to earn extra cash. This would not be official and nursery could not "arrange" it but get to know the staff and sound them out. Sometimes they are part-time have their own young children but can leave them with relatives whist they care for your own child.
WARNING - ENTERING DODGY TERRITORY HERE....
IMHO, from 6 years experience, nurseries are sometimes very quick to ring you and ask you to collect your child who "has a temperature" or "has refused all food and is lethargic". Sometimes I suspect that staffing numbers have dropped as a member of staff has gone home ill, they can't get any cover and so they have to reduce the number of children being cared for.
I have several times picked up a child who was being clasped in the arms of a nursery staff member only to have said child skip out to the car and ask for food or entertainment etc within 5 minutes. I have had this sort of thing confirmed "off the record" by an ex nursery worker who worked in a different nursery (although she seemed surprised that I had worked this out for myself).
Going forward, not relevant now, but something to bear in mind........
Make sure you tell nursery that it will take x amount of time from receiving a call from them to you arriving to pick up your child and they may not be so hasty in calling you. My problem was that I work literally on the doorstep of the nursery that I use and am there is therefore only a short delay before I collect my child.
I have on two occasions, I admit, given calpol at 7.00 am in the morning before dropping dc off at 8.30 and not told nursery, knowing that the staff will call me before giving dc any more so no risk of overdose (and you'd have to give a child a lot of calpol before it was dangerous). Both times dc was fine in my opiniopn even though temperature was slightly raised (cough/cold type bug both times) and I received no calls from nursery that day and picked up a happy child that afternoon.
You need to distinguish for yourself if your child has vomitted at home/had loose stools because of a bug or as a one-off for whatever reason as it is an instant 48hr ban for Diarrheoa or Vomitting.
Conjuntivitis - don't get me started on that one ! I even got called to pick up my oldest dc because of nits (we found 1 adult and a dozen or so eggs !) and could not drop them back off until treated. I did get back to work in the afternoon that day but only for a couple of hours.
In summary you do need back-up childcare if at all possible but it has to be searched for, it won't drop into your lap if GP's is not a possibility and it probably won't be cheap either.