Julie Bindel is not impressed in the Spectator:
Rosie Duffield’s opponents are intent on destroying her
https://archive.is/791Qp
As for being arrested by Nazis the fact is Rosie Duffield MP, and any women like her would have been arrested long before Izzard because the Nazis targeted left wing leaders, feminists and trade unionists earlier and in larger numbers than those notable only for being gay or cross dressing. Most SPD [Labour's sister party in Germany] delegates to Reichstag [equivalent to MPs] ended up in concentration camps unless they fled abroad, and the party was banned.
https://encyclopedia.ushmm.org/content/en/article/political-prisoners
Izzard would be way more likely to be targeted for his political beliefs than his clothing.
Pretty clear from history that everyone on Mumsnet feminist boards or in a left leaning feminist group would have been targeted by the Nazis. The Nazis only permitted groups under the control of the Reich's Women's Leader (Reichsfrauenführerin) which made sure women stopped thinking for themselves and obeyed Nazi doctrine.
Nazi policies criminalized women’s movements. They banned and condemned through propaganda and speeches the KPD (Marxist) and SPD (socialist) feminist groups first. Many of their members were arrested or assassinated, such as Libertas Schulze Boysen, Mildred Harnack-Fish, Hilde Meisel or 20-year-old Sophie Scholl. Others, like Helen Stocker, an advocate of women’s reproductive rights, Dora Schaul, a writer, or Lida Gustava Heymann and Anita Augspurg, who had appealed to the Bavarian Interior Minister for expulsion of Hitler, on grounds sedition, were forced to live in exile. Their property was confiscated by Hitler.
Then under a programme called Gleichgeschaltet (coordination), National Socialism gained control over the moderate women’s groups and then dismantled them. The majority of women associations, such as the BDF (Bund Deutscher Frauenverein), chose among themselves to disappear or disband. Nazi groups infiltrated the BDF’s subsidiaries. By May 1934, the BDF lost its power and influence. The only way to save the organization from becoming a puppet of the Nazi Regime was to voluntarily dissolve the organization. The BDF was established in 1894 and disbanded in 1933. The BDF’s extinction symbolized the “anti-climatic end” that Nazis brought upon the organized feminist movement in Germany in the mid 1930’s.
Only one women’s association persisted under the regime, Die Frau, or Woman, until 1944. It was placed under the guardianship of the Reich Minister of People’s Education and of Propaganda, Joseph Goebbels. Some old liberal feminists worked with international organizations, especially pacifistic groups, such as the Women’s International League for Peace and Freedom (WILPF) when World War II began. Others decided to wait until the Nazis fell, expecting, like most feminists in the Weimar Republic, that Hitler’s rule would be a short, ineffectual phase. Socialist and Communist feminists assumed a revolution would occur against Hitler, leading most to wait in silence or being jailed and killed. Though the Nazi regime failed as a government, it succeeded for generations in silencing German women and, therefore, putting a hole into the feminist movement for decades.
https://www.rgics.org/wp-content/uploads/Feminism-under-Fascism.pdf
Some of this seems a little familiar women's groups gradually being subverted, silenced while those who speak openly are crushed.