first things first - are you on medication for your sickness? It sounds like you should be!
I had an absolutely appalling pregnancy myself so I have some idea what you're going through so I'm just going to tell you what I did and then leave it up to you.
First I tried to ride it out, then I started taking odd days off, then I got some anti sickness medication (cant remember what now, little white pills twice a day) and that plus eating constantly kept me at work until I ended up taking a full month off for a different pregnancy complication.
The way I looked at it was pregnancy prepared me for being a mum, putting baby first meant putting me first, sod what was inconvenient for work and my colleagues, my baby was more important!
So, don't think about anybody else but yourself - what works best for you? if you can't physically be at work then dont be at work! spend the next few weeks/months doing as little as possible.
If you cant afford to not work, or think having to survive on SSP for god knows how long will cause more stress than the sickness then TALK to your work, they have to make reasonable adjustments to accommodate your pregnancy for example more breaks, permission to eat at your desk, changed work hours etc. (you can talk to the doctor about what they suggest)
It doesn't last forever (you'll have a baby eventually if nothing else) but it might last a while so do whatever it takes to survive it and DO NOT THINK ABOUT ANYONE ELSE BUT YOURSELF. You are growing a human - that shit is hard!