The only way I can think of to check if the ignition timing is being advanced is to connect up a Timing Light (Lamp) and watch the timing marks on the crankshaft front pulley and timing belt cover move further apart as you increase the engine speed from idle to about 3000 rpm (or whichever engine speed corresponds to full ignition advance).
As regards your engine running roughly etc. for the 1st 5 minutes and then smoothing out?
I'm assuming because of the age of your car that it's running on a carb.
In which case, I'd check out the choke operation. You
are using the choke when starting from cold? Also there's a mechanism built into the carb to pull off some choke as the throttle is opened, to prevent over-choking. With the air cleaner removed and engine stopped, pull out the choke knob (down beside the steering column, near the ignition switch?), look into the carb top, the choke flap should be fully closed. Push on the flap, it should be possible to partly open the choke flap (against light spring pressure).
Start the engine, depending on temperature, you may need to back off the choke a bit as soon as the engine starts (you don't need full choke unless it's freezing). Try operating the throttle or have someone press the accelerator pedal. As the engine is revved-up, the choke flap should partly open (as described above).
Try the above, if relevant and report back.
I don't think there's much that could be wrong with your ignition system, given that your engine runs well when it's warmed up. So I'd leave it alone for the moment and concentrate on the carb and choke first.
If you do decide to investigate the ignition system, check out my post above re:- the two sets of contact breaker points inside the distributor (unless you have electronic ignition) which alter the ignition timing depending on engine temperature. Might be related to your poor cold running? if the cold start c/b points were dirty/burnt/ incorrectly adjusted etc.
One word of caution. When removing the air cleaner body (not the top lid/cover), it's all too easy to drop one on the 8mm nuts (5mm thread) down the carb venturis. If the throttle is then opened, the nut can find it's way into one of the engine cylinders. (the nut can't get past the carb butterfly valves until the throttle is opened). So account for all 4 nuts!
Al.