It would have to be extremely rapid, going back to NBC training in RAFG and BAOR, when nerve agents were suspected or anticipated it was "be in time mask in 9", ie react PDFQ or die. At the first sign of nerve agent poisoning an autoject was used to inject atropine, each person carried 3 and it was a case of keep jabbing until the symptoms went away or you died. Everyone was trained annually to help himself and his mates.
The autojects, a spring loaded syringe, were kept at unit level and would have been issued on the outbreak of war or immediately beforehand. No great feat of admin involved really.
For civilian purposes in Israel I should imagine that army units kept sufficient stocks for all their personnel and civvy medical centres and pharmacies would carry stocks for the civpop.
Vaccination against Corona can be carried out in a more relaxed manner!