What is the opposite of 'lockdown'?

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

The family were relieved when the time came to let up.

The noun form lockdown has its counterpart in let-up — this hasn't lost its hyphen yet.

A pause or reduction in the intensity of something dangerous, difficult, or tiring

Ibid.

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.

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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.

' Normalcy or Normality ', I think , could be the appropriate word in this context

The opposite of lockdown is actually called freedom. Like in your sentence above it is better said like this: The family were relieved when the time came for freedom