You may have a problem with the battery/charging system or the starter
motor/solenoid/ignition switch. This is pretty basic stuff which any
mechanic or auto-electrician should be able to resolve.

Do the various warning lights turn very dim when you try to start? If
so, you have the first problem. It could be loose or dirty battery
terminals, a dying battery, loose alternator drive belt or faulty
alternator.

If the warning lights remain bright when cranking then the problem is
with the starter motor not trying to turn the engine over. More
likely, the motor is not being activated properly by the ignition
switch or the solenoid.

Did the work on the ignition switch pre-date the starting faults? Were
both the switch and steering column lock changed as a unit? If only
the lock was changed it's possible that the switch which is located
behind the lock may be faulty.

If you can access the starter motor and identify the solenoid which is
attached to it you will see a thick wire from the battery, a bunch of
other wires connected to its terminal and a single separate wire.

That single wire comes from the ignition switch and is only live when
you turn the ignition key during starting. Assuming that the battery
condition  has been eliminated, if that single wire goes live when it
should and the engine still doesn't turn over then you probably have a
faulty starter.

Not the kind of job you want to do in winter before work and not
really advisable if you are not at ease working in the engine bay but
maybe useful information when dealing with repairers.