Political party membership may be indirectly protected, insofar as party membership is a manifestation of a philosophical belief, but it is not a protected characteristic in itself. Political beliefs (e.g. in socialism) are capable of being protected characteristics, but not always. Nazism is not protected, because it is destructive of the rights of others and unworthy of respect in a democractic society.
Businesses can usually choose who they do business with
Assuming there is no other bad conduct going on (harassment, violence, discrimination, etc.), there is nothing to stop a shop or other business from excluding certain customers. This is the starting point.
Some businesses have positive legal duties to serve the public in general. Your water company can't cut off your water, regardless of your beliefs. Barristers must follow the "cab-rank rule" and take your case even if you are a horrible person. So let's assume we just have an ordinary private-sector business, such as a shop.
The protected characteristic of belief
Contrary to the term "protected class", which is from U.S. law, the U.K. uses the words "protected characteristic" to mean those personal characteristics named in the Equality Act 2010. These include age and race. They also include "religion or belief", a characteristic inherited from prior equality law. Any discrimination claim relating to the "No Nazis allowed" sign would have to hang on the "belief" arm.
Interpretation of "religion or belief" follows the European Convention on Human Rights, Article 9. In relation to a philosophical belief, the applicable test is "the Grainger test" formulated by the Employment Appeal Tribunal in 2009,
- The belief must be genuinely held.
- It must be a belief and not an opinion or viewpoint based on the present state of information available.
- It must be a belief as to a weighty and substantial aspect of human life and behaviour.
- It must attain a certain level of cogency, seriousness, cohesion and importance.
- It must be worthy of respect in a democratic society, be not incompatible with human dignity and not conflict with the fundamental rights of others.
Careful readers will notice that 2009 is before 2010; the statute law was different but the point was the same. Classic examples of beliefs meeting the test include vegetarianism, humanism, and pacifism.
There is no sharp line between a "philosophical" belief and a "political" one. Support of a political party is not covered (Grainger at 35), but there is room for belief based on a political philosophy. In this way, "left-wing democratic socialism" met the test in GMB v Henderson. On this basis, one might be tempted to conclude that "Nazism" or "fascism" would be similar, even if "membership in a Nazi party" was not.
However, Article 17 of ECHR does not protect totalitarian or Nazi beliefs. This is consistent in the jurisprudence of the Strasbourg court, and in U.K. cases on equality. It is the source of the fifth part of the Grainger test, via earlier cases in the House of Lords and before the European Court. For example, in Forstater v CGD Europe, the EAT held:
A philosophical belief would only be excluded for failing to satisfy Grainger V if it was the kind of belief the expression of which would be akin to Nazism or totalitarianism and thereby liable to be excluded from the protection of rights under Articles 9 and 10 of the European Convention of Human Rights (ECHR) by virtue of Article 17 thereof.
Employment is more protected
In Redfearn v UK , the ECtHR said a bus driver's rights had been violated following his dismissal for being a member of the British National Party. That was in relation to Article 11 (freedom of association) and his Article 9 claim was dismissed. The court noted that the BNP had not been banned in the U.K. and it was also looking at an employment rights issue rather than (as in the question) access to business premises; there is a general set of rights relating to unfair dismissal which were in play. It had not been proved that the driver was actually unfit to do the job.
So even though party membership is not protected under the Equality Act, people may still benefit from other legal protections. The case also shows the difficulties of inferring beliefs based on indicators like supporting a political party.
If the BNP were banned then it would be another matter, and U.K. law does allow the banning of extremist organisations. (Principally, under the Terrorism Act 2000, part II.) Some extreme-right groups are proscribed in this way, although there is no general restriction of fascism or Nazism.