First, it's often noted here that the medieval period lasted around a thousand years, so it's hard to generalize about the whole of it.

But, a very short answer would be that the blacksmith's wife woke him up, because as an apprentice he was a part of the blacksmith's household, and the blacksmith's wife would manage that household. The blacksmith and his wife acted in place of parents: they fed him, clothed him, made sure he went to church, and gave him a place to sleep. So, the apprentice likely wouldn't have to wake himself- someone in the household would do that.

Next, how early would the forge have to be lit? If it was burning charcoal, carrying a few embers from the kitchen fire over with some woodshavings would be enough to get blaze going within several minutes. For a coal fire, it could take longer, as coal is harder to start. Still, all in all, lighting the forge wouldn't take that long.

How to tell the time? If a rooster was crowing in the very dim light of dawn, the fire could be lit before there was enough light to get to work. But there were some people who very much wanted to know the time who didn't need to light forges: monks and nuns. Their days and nights were divided up with periodic sessions of prayer, and saying mass. They would be the first to develop mechanical clocks, in the 13th c. , and those clocks would have bells- small ones, at first, but then larger ones. Towns would also gain clocks, and bells.Those clocks were very inaccurate, having to be corrected with a sundial almost every day. But they were quickly set to the task of setting the work day. In 14th c. Brussels there was one bell at dawn, for kitchen maids, one for workers a little later. There would be a bell in the evening for the weavers and spinners to return home, then another a little later for tapestry makers, cobblers and fine metal workers. Eventually, workers would mistrust the accuracy of the clocks that only rang to set their working hours and were controlled by their masters. Fairer was to have the clocks ring at set intervals, first chiming for each hour, later every half-hour and finally every quarter hour. If the blacksmith was in hearing of a monastery or near a town, those bells could set the schedule for his shop.

David S Landes: Revolution in Time