As others have said, Stonewall is a charity. It is not a statutory body. It was formed to campaign against a law passed by Thatcher's government (known as Section 28) prohibiting local authorities from promoting homosexuality and prohibiting schools from promoting homosexuality as a family relationship.
After Section 28 was repealed, Stonewall moved into policy development and ran campaigns to equalise the age of consent (which used to be higher for homosexual relationships), end the ban on LGBT individuals in the armed forces, allow same sex couples to adopt children and/or undergo IVF treatment, and introduce civil partnerships.
They do receive some grant funding from the Scottish and Welsh governments, and their most recent accounts also show a grant from the Foreign & Commonwealth Development Office, but these are a relatively small part of their income.
For nearly 25 years they have been running a Diversity Champions programme, which has been joined by over 900 organisations. The programme is about addressing direct discrimination against LGBTQ+ individuals as well as more subtle forms of discrimination. It is through this scheme that they have been giving bad advice to employers.
To be fair to Stonewall, their advice was in line with that given by EHRC (the Equality and Human Rights Commission), which is a public body, funded by the government. The EHRC was involved in the Supreme Court case. They argued that the terms "women" and "man" in the Equality Act included trans women and trans men respectively and said that new legislation was needed to clear up the mess caused by that. The Supreme Court disagreed. An attempt to challenge the EHRC guidance through judicial review failed, with the judge saying that the case was "unarguable". I don't know the details of that case, but I wouldn't be surprised to find that the Supreme Court's judgement leaves that judge looking rather silly. However, Stonewall used that to justify their belief that the guidance they were giving was correct.
Having said that, Stonewall is very clearly captured by trans rights activists. Some of its founders have criticised its stance on these issues.
Stonewall cannot be banned or outlawed. However, it is clear that organisations should not automatically accept their advice as gospel.