We have a very similar situation at my shelter, we have customers "pre-adopt" prior to surgeries and then ask customers to pick up within two business days. Our current turnaround for spay and neuter surgeries is around 1-2 weeks but dentals take a bit longer.
We used to have staff calling adopters weekly but we inevitably had people who wouldn't get our message and would just see that we called and come rushing in only to be upset that their pet wasn't ready to go. It took a lot of staff time and seemed to just build frustration for customers that we'd be calling them with non-updates so we stopped that and now only call once their pet is ready to go or if there is some other sort of large update for them.
We use a waiver to help cover all of this, it outlines that they'll be contacted when the animal is able to go home but that we can't guarantee when that is. We recently updated it to list out what days they could pick up depending on the day the receive our call to help eliminate the guesswork of what we mean by a business day (since we operate different hours than most businesses). It also has a line that makes them aware that they can pre-authorize someone to pick up their pet for them and sign the remaining paperwork, and we have a spot at the bottom for them to provide that person's name and phone number. Since adding that we've had better luck with getting people to come in within our two business day limit.
The waiver is still being updated as we keep finding new tweaks to verbiage so I don't have one that I can share but these are the two parts of the waiver that seem to have helped us the most.
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