this happens more often than you think
and I am sorry that it has happened to you
the danger sign is - "we will do our Very Best ... " my ass
but now you have given notice that you need the time off
and so the 'sickie' will look very Odd Indeed
and is actionable ....like they can fire you
it happens a lot - Glasgow Children's refused a day off for a junior doctor to have hospital treatment [20 y ago] and he went around on the day - doing his stuff in the hospital commenting - isnt it lucky I only have cancer - I could have something really serious .....
I was refused time off - to do a course and get a diploma and bit of paper and the powers that be had said no because THEY hadnt done it and they were damned if I was gonna be the first before them....
it happens quite a lot - sorry
oh there was a time when my employers were gonna have the ass sued off them - and they refused me time off at work to talk to their defence lawyers. I contacted them and said I hope the plaintiff takes them to the cleaners and the employers' lawyers said yeah that is how we feel as well actually
I do NOT recommend throwing a sickie