Nokia has confirmed in a QA session over Nokia Conversations that Lumia Black will hit most of the Lumias by end of January. Though, this timeline will be valid for the unlocked carrier branding free Lumias, it seems. They still mention that timing of update depends on localized network carrier in reply to a query about update availability for Lumia 928.

While many Lumias have already benefited, the Nokia Black Update continues to roll out. Nearly all will have it by the end of January. Bear in mind that, for some models and regions, the timing of the update will be dependent on the localised network carrier.

So, looks like lumia 620 will also get the Lumia Black update in 1-2 days.