I'm a Swedish student wishing to undertake PhD studies in Britain. Apart from the fact that I'm not eligible for all funding, will having a non-UK degree count against me? I'm currently studying at one of the best universities in Sweden.

Having a non UK degree will not count against you. At least that is my experience. Both my bachelors and masters are non-UK and that didn't prevent me getting into a top English Uty for my MPhil. Having said that, it could be that my UK MPhil made it easier for me to get the PhD position that I am in right now. All the very best.

A Swedish degree should be fine. Have a look in the PhDs explained section of this site for more information and links. Also look in the funding section for information for international students.

swedish universities have a good reputation i think. for example, the karolinska institute.

I am currently doing a PhD in the UK and both my masters and my bachelors are also non-UK. Especially if your country/university has a good reputation, it will definitely not count against you.