Your GP will most likely be much more sympathetic than the receptionist. Just tell her it is a private matter you need to see the doctor about in person. They like to know in advance, but they don't have to know.
Don't worry about wasting the doctors time. This has made you ill with physical symptoms and s/he cannot treat them if they don't know the real cause.
If it helps, write down the symptoms it has caused on a list, all for them, from the visible ones like the eczema to the ones the GP can't see, like the crying, feeling sick, the lack of sleep, the rattiness with your family and the desire to quit your job and lack of concentration on anything other than worrying about work and your manager and the damage it has done to your confidence.
If you feel like crying in the doctors surgery then cry.
I hope the union are being more helpful to you right now than just filling in a form.
I don't know if they have advised you to keep a diary, but keep one.
Not just of the things your manager had done to bully you but off everything, the effects it has had on you and your family, everything.
Even something as simple as "felt weepy today, shouted at X and made him cry too. Feel too sick to eat lunch, went to bed with a headache. Can't face work tomorrow." Put down everything you think might be linked to the bullying, including bad dreams or reacting strongly to things that never used to upset you.
If you have to go through a grievance procedure or tribunal of any sort then that sort of think can help just as much as the actually things the manager has said and done directly.
And writing things down is a form of release, so it might help you feel better to get all that down on paper.
And also, if it helps, write a list of things you know you are good at and do well at work or at home etc, just to remind yourself. Bullying can make you feel worthless and you are not.
Your manager sounds like a total git. I hope that this is all resolved quickly for you.