if my T was running that late, and they do have emergencies, but he'd make sure i was given a heads up as a common courtesy to his client! having said that when you go back for your next appt. i'd share with him how his running that late puts you in a precarious situation at work unless he opts to reschedule you so you can leave. doesn't he have any evening hours? i'd ask about this too. a lot of people have some difficulty going for appts. during the work day.
does your workplace offer you personal days for this type of thing? do you have sick/vacation days you can use for the appt. day? these are just other suggestions in case you have not already considered them.
sounds like your T may be heavy booking which isn't fair to his clients that do show up. just thinking out loud...
the other thing that caught my eye...why do you feel guilty asking for this time off (so you can get better)? for us, it's the same as for some people that go see a specialist for a medical condition. it's important for our mental health. you don't need to feel guilty, you're helping yourself. that's a good thing.