With respect to lifting the lockdown of COVID-19, the news uses the word reopen:
[Merriam-Webster]
: to open again
// school reopens in September
// The Bay Area’s ink industry has been on hiatus since March, and plans to reopen tattoo shops this week have been paused again after a recent surge in coronavirus cases.
— Taylor Kate Brown, SFChronicle.com, "Bay Briefing: Raising the minimum wage amid the pandemic," 1 July 2020
In the example sentence in the question:
The family were relieved when the time came to reopen.
It would most likely be followed by some kind of noun, such as the economy or businesses.
If talking about social activities specifically, a more common word to pair with it is resume:
2 : to return to or begin (something) again after interruption
// She resumed her work.
// When official mourning was over, Soviet television resumed its normal pace.
— Bel Kaufman
With the example sentence again:
The family were relieved when the time came to resuming social activities.