What is the opposite of 'lockdown'?

Solution 1:

Some people in the UK and India have been using the word "unlockdown" as the opposite of lockdown.

For example, see this Sunday Times article on "The Great Unlockdown", referring to reversal of the lockdown that had been imposed due to COVID 19.

Solution 2:

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.

Solution 3:

Most recent uses of lockdown refer to the imposition of restriction of personal movement, and similar edicts.

I suggest that authorities let up on these restrictions (or even let up on them). It has a convenient mirroring on lock down.

informal (of something undesirable) become less intense.
Relax one's efforts.
let up on informal Treat in a more lenient manner.

— Oxford/Lexico