The Europol press releases says the arrested developer and employee were in Australia and Belgium, so it's probably law of those countries that's most relevant. The website had Australian phone numbers.
No, just a mutual aid request from an agency recognized under law enforcement treaty. There are many agreements for mutual recognition and execution of legal process across national boundaries (normally added as part of trade treaty negotiations but sometimes in things like extradition treaties). Lots of FBI raids outside the US are conducted this way - local police do the raid and have FBI 'observe' them in action. In this case, an EU police force and a 5-Eyes nation like Australia will be zero-friction recipient of assistance.
Presumably there would have to be some allegation of UK law-breaking in order to get a search warrant for properties in the UK.