-When did the siege begin?
-Who was controlling the Numidium? (If Anyone)
-Which Aldmeri Government was in Existence when it began?
For the first point, if we are to take the statement on face value, the siege "lasted from the Mythic until long into the Fifth". Time breaking and all of that jazz; timelines, world refusals, et cetera.
For a linear view of history? The Isles are conquered in 2E 896. With "Tiber Septim" presumably controlling Numidium, taking them from (I believe) the Second Aldmeri Dominion.
Someone else is probably better qualified to comment on that since recent depictions of Landfall have sort of soured me to the concept as a whole.
Have faith my friend. The answers are coming. And they will break your head more than the questions. Good times!
From the recent releases, we can be pretty sure that Numidium is anything but the salvation referred to by the prophet. Numidium killed the prophet; he refuted the prophet's message ("NO"). Nowhere in that scene does Numidium appear to be the salvation that the prophet is offering; if anything, Numidium is the threat from which the prophet intends to save the people from.
I would be very interested if Tiber somehow came with Numidium through the Dragon Break to attend Landfall in person. I would normally say that it is highly unlikely, and probably impossible. Then again, I would normally scoff at the notion of the Nerevarine turning up to the party as well, and we see how well that assumption would have worked out.