bilingual school Lausanne

These are quite different schools, they follow different programs and cater for different styles of families.
EBSR is a small, family-sized school (about 180 students, 1 class per grade) which mainly accepts local families (many of which are not “Swiss” but from mixed nationalities, often working in English) but also expats interested in bilingualism; they choose EBSR if they want their children to keep up the English curriculum and really learn French at the same time – or keep up the French curriculum and learn English at the same time.  EBSR has more teaching hours in the week to allow both languages to develop equally. This only makes sense if you stay in Lausanne for long enough.
They also have a high academic level compared to the local and international programs, a broader outlook and an individualized approach for core teachings. Most families choose EBSR because they have bright and curious children who thrive when they are challenged.
EBSR is located in a historical building in the middle of town, 5 mn walk from the train station, metro etc.; it is nice but slightly run-down, and they do not have great sports grounds, etc.
Ecole Nouvelle is friendly school where students are well cared for. It is much larger than EBSR. The grounds are very nice, in a quiet neighbourhood in Lausanne - a bit complicated to drive to (but they have a bus service). They cater for international and local families who want special attention for their children and no hassle. You would need to check what sort of bilingualism they offer before the IB programme they have for the last years.
Champittet has been developing since it was sold to Nord Anglia Education. The grounds in Pully near the lake are astounding and the environment is superb. Champittet used to cater for catholic families when it was founded more than 100 years ago, and had a reputation similar to Collège Saint-Maurice (its sister school) or Collège Saint-Michel in Fribourg. These days are over, and now Nord Anglia has implemented its teaching adapted to the 21st century, with a keen interest in new technologies. They mainly attract wealthy Swiss and expat families. On what concerns bilingualism, you would need to check what sort of bilingualism they offer. Apparently they get good results.
ISL has been growing a lot since they built their new campus in Le Mont, where they offer very modern premises and lots of sports grounds. The school is an IB school and follows the PYP ad the MYP. These programmes are not designed for a fully bilingual education; however, most IB schools offer lessons in other languages. Whether the students will indeed become bilingual is a matter you must inquire upon. Since ISL is more or less an “Expat Bubble”, families mostly stay among themselves and do not “really” learn French. Further, since many students are very international, English is not their mother tongue and the general level of English is not the one of natives.
Choosing a school is not an easy task. You might want to get interviews with the heads in order to make up your mind. If you are really interested in bilingualism, you need to check what the schools really offer and what their results are.
Good Luck!

Thanks a lot . It is very detailed and helpful.