I had a work visa that would expire in December, but I received a new study permit. However, this visa came late and college deferred to me for January.
You should apply to change status as a visitor. That will legally keep you in status between your work permit expiring and your study permit starting.
Apply just before your work permit expires. That way, you'll be in implied status till you get a decision - and your studies may well begin before you get a decision so you won't even care if it's denied.