Is the German embassy in London bound by the Equality Act 2010 and cabinet office guidance on medical mask wearing exemptions? The website explicitly state that in order to enter the Passport section of the embassy it is "required to wear a protective face covering (without valve)". Further it mentions that "In order to claim an exemption you will need to present a corresponding medical certificate in your name." So the general question would be whether the German embassy in London is bound by German law or by British law?
Laws of the host country do apply in foreign embassies, but the authorities of the host country may only enforce laws in a foreign embassy by invitation of the foreign diplomatic delegation.
So the general question would be whether the German embassy in London is bound by German law or by British law?
It is subject to both, but the embassy can decide to what extent British law applies.
In the case of a mask mandate imposed by British authorities, the embassy may require its occupants and visitors to comply with the conditions imposed under British law. It could also refrain from imposing such a requirement. In fact, unless German law prohibits a mask requirement, it can impose a mask requirement on its own initiative even if there is none imposed by British law nor German law.
Is the German embassy London bound by the Equality Act 2010 and cabinet office guidance on medical mask wearing exemptions?
It is not bound, but it may comply voluntarily.
Embassies are special
Embassies are special in that by default not the laws of the host country apply, but the laws of the country that has them regulate their operations. That's more a practical thing: Law enforcement of the host can't do anything in the embassy that isn't allowed to happen by the ambassador.
In the German embassy, no English law can be enforced without the consent of the ambassador.
Any office is free to put up requirements above the ones required by law. If UK law requires that you need to wear a mask when entering an office like the German embassy, the German embassy must require you to wear a mask. If UK law doesn't require it, then the German embassy is allowed to require a mask, and to require proof of an exemption if you don't want to wear one.
The German embassy in London would also be required to follow German law, but not because it's the German law, but because their paymaster (German government) tells them so.
But the question has nothing to do with being a German embassy, it would apply to any office in the UK. Any office can go beyond the legal requirements.