In relation to two of those searches, Police relied on the provisions of section 18 of the Search and Surveillance Act 2012. This gives Police the power to enter and search a place where they suspect an offence against the Arms Act 1983 has been committed and the person is incapable of having proper control of the firearm(s) or may kill or cause injury to someone. The Authority found in these two cases those circumstances did not exist when the searches were conducted therefore, they were unlawful.
The third case related to Police entering a house where they relied on section 14 of the Search and Surveillance Act 2012. This provides the power to enter and search without warrant in circumstances where it is believed an offence is being or is about to be committed, or there is risk to life or safety that requires an emergency response. Again, the Authority found those circumstances did not exist at the time therefore the entry into the house was unlawful.
The full report is here, and the police behaviour is... dubious (and in the third case, outrageous). Obviously, these people were a legitimate subject of concern. But the police could not even be bothered building a tissue of a case to justify a search. As a result of that, they've been forced to apologise, and will no doubt be paying compensation to these nazis. If you're upset about that, you should direct your ire at the useless police who didn't do their job properly, rather than their (fairly repugnant) victims.
Meanwhile, this obviously raises the question of how else police use their warrantless powers, and what further scrutiny we should subject them to. Every use of a warrantless power must be internally reported, and agencies are required to issue statistics on their use, but clearly we need someone reviewing actual reports for abuse and dubious patterns. Because the police clearly cannot be trusted to use these powers appropriately, and we should not let them endanger prosecutions with this lazy approach.