Looking back at my rulings, it's not really clear if I went with healing happening as of the game date, or the real-world date. It's clearer on ordering items.
Here are what I'll use as my canonical Felltower rulings from now on:
Healing occurs on the game day when it is applied. For spells known, that can be as little time as it actually takes to gather the energy and cast it. If the healing requires an external source (elixir, hireling NPC, scroll) or the church, the healing occurs on the game day when it is available and used. Date any extended recovery time from that point. This only applies for multi-session delves; if healing is needed after a "normal" delve but not applied until a later date, use the actual real-world date or in-game date, whichever is later, for the start of the healing process.
Special orders, attempts to hire hirelings, sage research arrangements, etc. occur on the real world date or the game date when the order is made - whichevr is later. They cannot be back-dated to the game day or an earlier period of time.
Putting it together
For example - the delvers go into the dungeon on November 22nd, but it takes 3 sessions to finish the delve. The real-world date is December 21st when they finish the delve . . . but in game, they've returned to town very late on November 22nd. Healing takes place on November 22nd (if they have the spell handy), on the 23rd otherwise (if the spell needs to be prayed for, or a scroll purchased, potion purchased, or the church contacted and healing donated for.)