I’m no hr expert but I believe If you have worked there for over 3 months you can only be dismissed instantly for gross misconduct. Otherwise they have to follow due process, verbal warning then written then dismissal. Your employer should be accomadating with requests for doctors appointments, if you show regular sickness then it can be reviewed but is not always a dismissal. As for working your notice because they kicked you out I believe they should pay you. Speak to acas but sounds like unfair dismissal to me