A British person with a German passport is also a German citizen. So they can't not help them if they have the means.

I don't know about the Brits. The Germans have some tight rules about dual citizenship. It will be an exceedingly low number of people who qualify legally on both sides for dual citizenship of these two countries.

As the revolutionary guard troops surround Dubai, this probably would not matter. They would try to get people out and I'm sure both Brits and Germans would evacuate each other's citizens if it meant death if they didn't. But if the situation allowed for more thorough investigation, I probably wouldn't mention my other citizenship to the Germans at least.

Many folks of Turkish descent but born in Germany used to take up German citizenship and had to renounce the Turkish one in the process. But they were able to get it back once the German passport was done, which could be grounds for revocation by the Germans if they knew about it. In Japan it's even harder to have two passports; kids of mixed couples often have two passports but they're not supposed to from the Japanese side. So they wouldn't make this known to the Japanese embassy who would care about this if the revolutionary guards were still far enough away to have a closer look.

I would not be surprised if there was a bit of horse trading going on between the embassies. So if a dual citizenship holder was rescued unconscious in a drone attack, it's probably the first delegation on the scene who takes charge of the case. Until the victim regains consciousness and possibly decides something else - if they can without causing more trouble for themselves.

I have no experience with this so my guess is really just that. If two consular outfits arrive on the scene to help our unconscious victim at the same time, they would probably try to figure out: what's the country of residence for the victim? If it's not a third country, residencistan trumps the other one. If it is a third country, then where are the next of kin. If there are none, probably country of birth. If that doesn't help they do a game of paper, rock, scissors.