Childcare for children aged 0-3 falls under the auspices of the Ministry of the Economy, which supervises subsidized daycare centers through organizations such as Naamat, WIZO, or Emunah, and private initiatives, which are not subject to ministry regulations, although many do have the oversight of private regulatory organizations.
Daycare staff and nannies are not trained by default in allergy care so you will need to take the initiative to ensure that they are trained properly. You should also ask them for a written agreement stating that they are responsible for administering medication and calling an ambulance. You, in turn, will be required to give written permission to do so and commit to paying for ambulance costs.
Allergic children in government-regulated daycares may be eligible for two hours daily of private supervision by a personal Sayaat. This is provided by the Ministry of Social Welfare (Misrad HaAvoda, HaRevaha VeHashirutim HaSotzialim) even for families who would not ordinarily qualify to register with the Welfare Ministry. The Sayaot provided by the Welfare Ministry have a strictly supervisory role and are not authorized to administer EpiPens or other emergency medication. Thus, it is upon the parent to find out who will do so, and to train them accordingly.
Many Israeli childcare providers are happy to adapt their menus to meet the needs of an allergic child, although this is more easily accomplished in private frameworks, and it is important to check if allergens will be totally banned and removed from the premises or if the child will simply be given a special meal. Welfare Ministry Sayaot are rarely authorized for private frameworks.
For children who have severe allergies to multiple foods or airborne allergies, it may be difficult to find a childcare center that is up to the challenge. This is also presently under review in the Knesset. In the meantime, for children under age 3, it may be easiest to bring a private babysitter into your home, rather than sending the child to group care.
A few initiatives have been taken to open allergy-friendly childcare centers in Israel. Some leads can be found at: https://web.archive.org/web/20110305031715/http://www.foodallergy.co.il/P3052.