Not to sound like a smart
but the electrical diagram in the back of haynes/chilton book is marvelous thing. But to get started solving your problem, let me sum up your post, just to make sure I understand what you have.
1) a pushbutton hooked, presumably, to the starter relay
and
2) a single switch to turn everything on/off
So your process is to flip the switch to 'on' and press the button to start.
Is all this correct?
Now, I'll drop some notes about typical old style key switch electrical systems. There are generally 4 main circuits tied into the key switch system:
1) IGN1 - usually includes the primary things that allow the enigine to run, such as the coil and fuel injector relays/fuel pump, if applicable.
2) IGN2 - usually other things of high importance to driving, but not required to run the engine, such as wipers, turn signals and relays and AC/heater blower
3) ACC - usually stuff such as the radio, cig lighter and such.
4) Start - duh! the momentary switch to engage the starter to start the engine.
So, in order to remove the key switch entirely, you have to rewire anything there that you intend to use, with IGN1 and Start being the 2 most critical ones. Keep in mind, that the switch your using has to be rated for the amount of current that you will be putting through it. This is one reason why most competition rigs you see with a push button start have a whole panel of switch. Well, that and safety.
Something else to consider in removing the key switch, the key switch also controls the steering wheel lock, so, if you don't want to turn the key at all, but want to turn, you'll have to remove that mechanism as well, which will leave your steering wheel permanently unlocked.
Now, all this information is relevant to the older vehicles that don't have a computer control of much. All bets are off if the vehicle is OBDII.