For a city, the notice must be sent to the Mayor. If you miss the deadline by even a day, your claim expires. Banks and safe deposit boxes. Fortunately, banks do not fail very often. If a bank is not doing well, the government agency that regulates it usually causes a failing bank to be bought out or taken over by another, healthy, bank.
Customers never lose any access to their safe deposit boxes.
If the agency does not find another bank willing to take the failed bank over, the agency may temporarily keep the bank open itself. I guess it depends on what one means by being away for an extended time. If the customer was away for years, and was so far away and isolated the customer did not hear of the bank's failure, there could be a problem. But if one left a mailing address at the bank, as one should, the customer should receive a mailed notice. I think that realistically there is very little real risk in this situation.