I recall that one of the issues about judges, police etc belonging to organisations such as the BNP was that it could affect their ability to carry out their duties in a fair and responsible manner.
I'd better be careful in how I word this, but even at the lowest level, of, say, a police constable or JP, there is an amount of leeway as to how diligently a matter can be pursued. If, for example, a BNP-supporting police officer were called to investigate a crime with a black victim, they may allocate less time and effort than to a similar case involving a white victim. Similarly, a JP with far-left leanings may view certain people up before them with more or less sympathy than others. If you work in a post where your political views may affect your judgement, it could be tricky. People in such positions, with such leanings, could find themselves going to the other extreme - a bit like if you're a teacher and one of you own children is in your class, you have to go harder on them so as to pre-empt any accusations of favouritism (Malcolm may disagree on this point
If you're a member of, say, the Conservative party, you'd reasonably have no objection to the Labour party's right to exist, campaign, and if elected, to form a Government. Same goes for any mainstream party - you may disagree with other parties' views, but you acknowledge their right to express them. On the other hand, if (God forbid) the BNP ever got elected, you can bet your bottom dollar that any dissenting views would be made illegal. That's the difference. They are intolerant of other views. The mainstream are tolerant. Why do we allow the BNP to exist? Because we're better than them.
So, while I have no objection to people expressing whatever views they may have (provided it is within the law), I do have an issue with people of extreme views holding any office where those views may compromise their ability or willingness to do their job properly and fairly.