Thanks for the replies! I (and she) are well aware they legally have to let me go, but I was trying to be fair by offering a compromise and swap my shift so that the appt would then be in my time! But no, that doesn't seem good enough!
Whatever! Their problem now. I'll be leaving early on wednesday, and it is a long appt, so won't be back!
Had a tearful day off today, and nothing seems to be lifting my mood! Shouted at DS this morning and he shouted back at me, then we both burst in to tears. Arghhhh. I'm just waiting for him to get home with DP so we can discuss things calmly.
All because yet again he made us late for school 
. When I am off work, I get up at 0810 to leave the house at 0835. In that time I get dressed and dry DS's hair (longish hair needing daily washing and drying). DS likes to get up early to watch TV. He gets up at 7. I ask him to make sure he has breakfast, and to get in the shower no later than 8am. When he gets out the shower he needs to start getting dressed no later than 8.20. The only other thing he has to do in the morning is make his bed and open his curtains.
Yet he is incapable of this. Apart from his shower, he frequently hasn't had his breakfast, or got dressed, or is half dressed, or tells me he needed xyz for school that day. He turns what should be a simple task in to a frenzied stressful time of me nagging him and running round after him, and us ending up leaving late. He does the same to DP too. When he is at my mums in the morning, he doesn't do it because he isn't allowed to watch TV until he is ready. I don't want him getting dressed too early because our dog is very hairy (long haired golden lab) and he ends up covered in dog hair because he is incapable of sitting neatly on the sofa. He plays with the dog, sits on the floor, etc etc. So it wouldn't work well that way round for us.
I have the same problems at the other end of the day when he gets in. He has to put his coat and shoes away, bookbag away etc and take his uniform off so it doesn't get covered by the dogs hair. Yet he has to be nagged again to do this.
He is 10, and at 10 I was taking myself on the bus to and from school fgs! He is perfectly capable of doing all of the above. But it is becoming more frequent for him to not do it. He has had the same routine in the mornings me and DP take him to school for the last 2 years. We take him 2/3 times a week. His dad takes him once a week, and my mum 2/3 times a week. Depends on if I'm on an early shift.
So this morning when he turned up in my bedroom at 8.30 to have his hair dried, with just school pants and shirt on (untucked, and buttons not fully fastened) I just lost my rag. He had been reminded 10 minutes before to get fully dressed.
So, TV is going to be banned in the morning. I don't know if this will work, because knowing DS he probably just won't get up as early, and will find another distraction instead .
God help him next year when he will have to get ready and walk to secondary school 2/3 times a week when I won't be here!
Anyone got any good tips on how they got their children to take responsibility for getting themselves sorted in a morning??!!!