The 29th of June I brought home a happy and healthy little puppy who had had it's first vaccination on the 26th.
The 10th of July I took the puppy (still happy and healthy) for her 2nd vaccination. She was fine that day.
The next day the puppy was lethargic and only are a tiny amount of food in the am and refused food after that.
The puppy did not eat again until Monday or Tuesday and in the meantime I had to force feed with a syringe.
She improved slightly since Tuesday and does eat a tiny amount of food but has no personality like before the vac. She comes in the house goes straight to her cage (or a corner if you close her cage) and sleeps. No interest in playing at all. She won't come to you when called, ignores her favourite toy, and has no motivation for lamb treats like she used to.
She went to the vets at the weekend and he said it wasn't the vaccine and didn't seem bothered. Just said do a blood sample, but as we wouldn't get the results for 3 days we refused. As she improved a little we didn't take her back but are now considering another vet as she isn't right.

This morning she had dark diarrhoea in her cage with what looks like couscous in it. I'm assuming these would be worms. If it is worms, could this be the cause of the problem? And what wormer would be best to use in a 800g puppy who isn't especially well? She had been wormed with 1 dose of Panacur on Monday as I didn't want to do the 3 day course incase she took a turn for the worse...

Is it a Papillon (judging by your username)? A lot of Papillon breeders even ask their vet for a smaller dose of vaccine as they can be sensitive. I've never done that, but I have learnt that they DO get unwell after being vaccinated the first time, usually with a lot of pain all over the body, sickness etc, and so now, on my vet's advice, we dose the pups with Metacam an hour or two before being vaccinated. This seems to be working -and a year later for the first booster they are fine without Metacam. I do however have one, my smallest adult at just 2 kg, who developed serious pain for a week after every vaccination even when boostered at adult age, so she is my only dog in well over 30 years that now is not vaccinated every year.

As for worms, personally I prefer Drontal puppy suspension as it is just one single dose every 2 weeks up to 12 weeks of age.

I think you need to get her back to a vet sharpish.  A different vet, though, yes - reactions can and do happen (my first pup's first shot could have killed him if we hadn't take him back in that day) and such a blaze attitude is concerning.

Diarrhoea in a very young pup is also very concerning, they do not have a lot of reserves (especially in such a tiny pup) and can do downhill very quickly so she needs attention now.  The vets will be able to advise you on worming if it is that but this 'couscous' thing sounds odd - could be worms but I think you should get it checked out.  Take a sample with you if you can.

>This morning she had dark diarrhoea in her cage

This would have me very worried.  If it's dark then I would be concerned in case it's colitis.  Vet!  Now!