Not far into future at all. I've established they could conceivably beat the Arks to Andromeda (which are going to take at least 240 years if they use Reaper tech), and after the arks arrive, the chances of contact go up with every passing year. I can't see a good reason to explain why contact couldn't or wouldn't be made relatively soon.
But, like I said above, "conceivably" isn't enough to cause a problem with the ME:A scenario. Unless it's "inevitable" rather than "conceivable" that the MW will launch an expedition which either beats the Arks to Andromeda or arrives almost simultaneously, this just means just means that something which might have happened didn't actually happen. That's a nonissue, an obvious nonissue.
As for "relatively soon," depends on what you mean. Bio could advance the calendar between installments the way TES games do, but that isn't their typical MO.
Sure, the can could be kicked down the road for a while, but at some point Bioware are going to have to decide whether to deal with the endings and risk more rage or face a growing number of fans asking questions. I say bite the bullet and get it over with.
Bio can kick this can forever, if they feel like it. There's no reason to ever advance the calendar to a year when contact with the MW is necessary to preserve the plausibility of the setting, unless Bio feels like it.
It's sounding like you're casting around for some reason for Bio to be forced to do the thing that you wanted them to do all along.