What drives N1 during a jet engine start?
N1 is driven by the starter flow through windmilling. N2 is started through a gearbox as described in this answer, driven by an air starter or by an electric motor. This causes air to start flowing through the engine, and the LP turbine blades pick up energy from this airflow.



Above graph is from The Jet Engine by Rolls Royce, and shows that the LP spool (N1) starts to turn in the dry cranking (starting) phase, after N2 has been accelerating for a while. The speed with which N1 spools up depends on the engine type, but even the large inertia of a helicopter rotor is slowly starting to be driven around by the starter flow.