I studied with rabbi and got very good insight from him regarding OT and why Jesus is never acceptable to Judaism as a Messiah. Educated Jews find NT laughable as it is far from the teaching of the Judaism and its core beliefs and values.
Couple days ago I have discussed on different thread regarding literacy of Jews in the time of Jesus. 90% of them in his time were illiterate and they had to depend on oral tradition and memory to recite their beliefs and teaching. Only handful of priests were educated and had access to various reading material. It was time before print press and copies of the manuscripts were expensive to do so. Literacy among Jews before the Renaissance was still bit higher than for majority of Europeans, but still two thirds were illiterate. So I do not think the masses during Jesus time were well educated in the OT or had access to them.
Split between Judaism and Christianity was started by apostle Paul to make old Jewish beliefs obsolete and new religion based on Jesus death desirable. In the 1st century, this split did not exist and was completed by 100 AD, about three generations after death of Jesus.