What is your status in Israel? Are you a temporary or permanent resident of Israel and are registered at the Belgium Embassy in Tel Aviv or the Consulate in Jerusalem?
In this case your passport only needs to be valid on arrival and you can travel as planned and renew the passport after returning to Israel. It takes 3 - 4 weeks as an average.
http://www.timaticweb.com/cgi-bin/tim_we… ..
In Dutch:
http://www.diplomatie.be/telaviv/default… ..
In French:
http://www.diplomatie.be/telaviv/default… ..
Should you just be on a tourist status in Israel and stay visa exempted I am afraid that you have to change your travel plans. In this case you would need a passport which is valid for at least six month beyond your intended stay. You would need to go back to Belgium from London and apply for the passport through the commune/gemeente which is your main place of residence. It usually takes 5 working days to produce a new passport at the regular fare of €71 plus communal fee if applicable. The actual time depends on the pick-up dates of the community at the passport production facility. You also may get it within 2 days on the speed-procedure at a fee of €240.
http://www.diplomatie.be/nl/pdf/prijsVan… ..