We put our kids into a bilingual Montessori school. At the end of 12 months, after plenty of angst, I wondered if the children would have adjusted better with full immersion in the local school...did we make the right choice and was the cost justified (our kids tuition is part of my husband's salary package arrangement)...?
In our case, the bilingual school offered 50/50 with a native german and native english speakers. And the assisting staff and children speaking dialect in-between...However, neither teacher had ever had a child who started with zero German...and it took at least 6 months of 'trusting' that the children were learning stuff about the language, before the evidence started to appear...
I am *convinced* that our children would have learnt the language faster, and made connections in our local community through being at the local school (and financially it also would have made sense)...taking the bilingual option was a 'soft' option for us because it meant we have been able to deal with all issues arising with the teachers in our home language. Also, our middle child has specific medical needs which affect him daily in the classroom, and the bilingual school accommodated this as well as providing him with meals to match his dietary requirements - I'm not sure I could have handled this with a local school across the language barrier...oh, yes, and my husband's a teacher, so the kids are at his school, which has made it much easier for me as in an emergency he is usually 'on site' already...
I will definitely try to reassure you YES YES YES! - your kids will learn the language, and 1000 other things about people and culture, and life and independence, that they would not learn back 'ome...
Being in the local school at age 7 and 4-5 is an excellent time to 'transition' - they will make adjustments, and the main thing is to not freak out about the 'academics' but recognise that the schooling system at the primary level is as much about broad experience and immersion in learning, as it is about specific academic outcomes...you won't be able to measure how they are going academically for a while, until they start to find their feet in their new school language - but I'm sure they will be fine.
Our daughter came at age 9 with very advanced English language skills. For the entire first year she refused to speak German at all...and she was assessed at the beginning of this year within the class - the teachers set her a year 3 level German exam (she's technically year '5') - but she actually did very well - so the general impression is that kids somehow traverse the language barrier and their strengths in English can translate to strengths in German also - she's also very 'switched on' to French...
However, other than the occasional outburst from our middle child, and our toddler who is babbling both language together...we don't hear what our children can do in German, because we don't speak German at home - but when they are socially with friends, or in the community, or at school, we rely on their teachers to tell us that is going OK...
We will stick with the Montessori because that is our choice for 'learning style' and 'curriculum' - and we are happy with the quality of the French and German, as they have a good team of mostly native-speakers...but I think adapting is more about the family/parent attitude, immersing in the local community and participating in the social life, than the actual technicalities of school...