Peter,
The next lead which matches your Engagement Campaign's trigger will enter the Stream. Now, I don't know what trigger that is and when a lead would match that trigger, so there's no way to tell WHEN this will happen. It's fine if it takes several days assuming that you only want leads matching that criteria to enter the Stream.
It's better to setup some test leads and then manually trigger them to enter the Stream. Once you are certain that this is the right set and Cadence, you can then migrate your leads out of the OLD system and into this new system. You can also set the Entry Trigger to add leads to the first Stream.
As for the other Programs you are running, they can pull in a lead at any time as well (depending on triggers and filters), so it is entirely possible for a lead to enter your Engagement Stream AND several Programs>Campaigns. That's how it should be.
Of course, if you don't want certain leads to enter all Programs or any Program, then you need to add exclusion filters like
Added to SFDC Campaign IS "X"
Member of Program IS NOT "Y"
so that any current member or previous member of Program Y is excluded from the Stream.
Take a look at my
Engagement System summary because I think you've got some of the terms confused. A Transition Rule is only between Streams and doesn't work on external campaigns/programs.