Saul - first off, welcome to the forums! I'm looking forward to hearing more about your experiences and ideas.
As to the non-responders,...
Maybe we can get a better picture as this progresses, but I'm going to say patience, as my first bit of advice

...
If I'm understanding right, these Saturday kids come once a week for a solid block of time and then go away for days without English input. They have little need for English in their daily lives and some might not really understand why they go to English playtime every Saturday.
I'd say keep activities and response simple at first and building. Small successes for students are very encouraging and they want to impress the teacher. Praise individuals in front of the group and you'll start to see other kids trying to show off by copying, ie speaking English. Eventually you reach a snowball level where all of that time you spend with them will start to develop into more language.
It also sounds like your regular kids and after school kids don't mix. If that's the case, that's too bad. The regular kids probably learn a lot from each other and I've found that having my own kids in kindergarten classes is a real plus. The kids that understand what to do respond. Then the other students learn from them what it is they are supposed to be doing.
Normally my classes with older students progress slowly over time and after about a year and a half I can't really put a new student in with a running group. However, with kids 5 and under, it's better for them to go in with a long standing group and learn from the other kids. So, if you can mix some of these students with students that have been there a while, it might help.
Anyway, can you tell us what sort of activities and content you are using and maybe we can offer some advice on that.
Additionally here's a post that sort of sums up my thoughts on what makes for good practice with young learners or how to reach them:
http://www.mes-english.com/forums/viewtopic.php?t=10