At the moment, nobody knows.
The former Chief Medical Officer (the man who proposed the border closure last year) was interviewed on TV a few days ago and made the point that it’s exceptionally difficult to predict more than about a month ahead in this situation. There could be another covid variant around the corner, or the vaccine rollouts around the world could speed up. Both situations would radically change the border closure situation, as you can probably imagine.
He indicated, in the same interview, that he doesn’t see much in the way of the border opening until 2022, but that that wasn’t a guarantee.
Yes, there are plans well and truly afoot to start vaccinating people here early next month and to ramp things up by March - I think the suggestion is that there could be as many as six million people vaccinated by the end of March (we have a population north of 25 million, so 6 million is a good number but still a way from the project being completed) - but that’s only half of the issue. If we’re going to open the border to somewhere (or everywhere) else, we’d need to know that the people coming in aren’t likely to spread the virus themselves.
To put it in simplest terms, family members overseas aren’t expecting to be able to visit before 2022. If the situation changes, they’ll be happy, but they don’t expect it to.