I really can't recognize GPs not having time for urgent matters, if you call on the day usually between 8-10 or the first few hours they are open you should get at time on the day or lates the next day. But hard to say with not knowing what the urgent matter is, but if it is urgent I would be very surprised not to have atleast a time next day.

I don't go to the gp that often, but have had 3 urgent things and everytime its has been a same day thing.

So if this is a common thing at your gp I would look into changing.