I think you unwittingly truncated that verse and cut off one line. The fuller verse is
Footfalls echo in the memory
Down the passage which we did not take
Towards the door we never opened
Into the rose-garden. My words echo
Thus, in your mind.
Take notice of the key phrases here: "the passage we did not take"; "the door we never opened". We (humanity) didn't take that passage to the Garden of Eden. The road to it has always been a one-way street. We, or our original ancestors, the primordial humans, first representatives of the human race, were expelled from it. Thus, we, never opened the door to the Garden of Eden. But the garden exists in our collective memory, through time and throughout time.
Emphasis should also be placed upon what precedes this verse:
What might have been and what has been
Point to one end, which is always present.
One popular interpretation of these lines as well as the temporal theme of the poem is that it is thus evidenced by these lines how Eliot considers only the present, not the future or past, really matters. The present moment, some would argue, is the only real moment in Eliot's poem, as the past cannot be changed and the future has yet to happen and is forever decidedly unknown. I disagree. The past and the future both point to the present and manifest through the present. In a sense they are part of the present. The past and the future fold onto the present.
You ask "how, when and for what purpose we moved from "time" to "rose-garden"". It is a rather seamless and smooth flow. Consider the Garden of Eden, there is a perpetual "what might have been" in front of the humankind, the ultimate mystery: What might have the humankind been like had Adam and Eve not eaten the forbidden fruit?
We can't answer that. No mortal soul can. That is why prior to those two lines Eliot wrote:
What might have been is an abstraction
Remaining a perpetual possibility
Only in a world of speculation.
We mortals can only speculate. We sinned, and we were punished. The other path has always and will always remain a possibility, an abstraction, a blur. So to answer your first question, I'd say Eliot is not moving per se from a discussion of temporality to a random rose garden, but rather the Garden is the thing he is trying to get to by leading with a brief discussion of temporality.
"We" refers to humanity. Eliot is taking all of us on a virtual/impossible tour in the Garden of Eden. Keep in mind that the poem had been partially inspired by a real-life manor house in Gloucestershire, England. Eliot is giving us spatial images the same way he handles time: he brings the imaginary Garden of Eden onto the destroyed and deserted Burnt Norton estate. Through his poetic language we see spaces, temporalities, and possibilities all overlap. That is why although the place in real life was burned and deserted -- a metaphorical reference to the Garden of Eden being destroyed and abandoned -- we are taken to a beautiful garden full of joy, guided by "the bird".
The bird here could be a biblical reference to the birds in the Garden of Eden:
Then God said, “Let us make mankind in our image, in our likeness, so that they may rule over the fish in the sea and the birds in the sky, over the livestock and all the wild animals,[a] and over all the creatures that move along the ground.” (Genesis 1:26)
God blessed them and said to them, “Be fruitful and increase in number; fill the earth and subdue it. Rule over the fish in the sea and the birds in the sky and over every living creature that moves on the ground.” (Genesis 1:28)
Hans Christian Andersen wrote of such a bird:
Beneath the tree of knowledge in the garden of paradise stood a rosebush. And here, in the first rose, a bird was born. His plumage was beautiful, his song glorious, and his flight was like the flashing of light. But when Eve plucked the fruit of the tree of knowledge, and she and Adam were driven from paradise, a spark fell from the flaming sword of the angel into the nest of the bird and set it afire. The bird perished in the flames, but from the red egg in the nest there flew a new bird, the only one of its kind, the one solitary phoenix bird.
Phoenix bird! Don't you know him? The bird of paradise, the holy swan of song? (The Phoenix Bird/Fugl Føniks by HC Andersen)
But it could also be a MacGuffin. Why a bird? Because they are seen as cheerful.