It's an emergency. Doesn't matter what. Give him what he wants unless it's going to cause big boom.
While this is true, one HAS to temper this "move everything out of the way for me NOW" attitude, with understanding that it might take time to do that.
Years ago, I heard something on the ATC freqs that caught my attention. There was an IFE (inflight emergency) in progress. In that instance, an aircraft declared an emergency in Class B airspace. The weather was broken CB with rain, enough-so that IFR aircraft were being vectored to instrument final, with no visual approaches being issued. The aircraft with the IFE (rough-running engine on a Cessna 421) couldn't understand why he wasn't getting immediate vectors to final to land, and whyhe was told to "stand-by" by ATC. He raised hell a few times about it to ATC. But there seemed to be, from my perspective, a few things he failed to understand about ATC and how they work. Now, I fully understand that anytime someone declares an IFE, they expect, and should receive, priority handling to the maximum extent possible. Keep in mind the last part of that sentence: "to the maximum extent possible." When someone has an IFE, their only worry is (rightfully) their aircraft and themselves, but they've got to understand why somethings may happen that may not initially go their way. When an IFE gets declared to ATC, the controller's entire scope doesn't come to a grinding halt. He still has those (possibly) 10+ MTI "blips" to sequence and separate. Once someone declares an IFE, chances are, they've now thrown a monkey wrench into the controllers sequence/separation plan; they're a pain-in-the-ass now for all intents and purposes. He still has his other traffic to work while he works to prioritize the IFE, and may very well have to have the IFE "stand by" while he coordinates with other sectors/tower for the IFE, etc; this workload being possibly multiplied ten-fold if the WX is actual IFR due to sequencing needs and the lack of visual approaches/separation available. Often times, an IFE may just have to recover within the flow that's already in place. It's a crap-sandwitch, I know, but you play the cards you're dealt.
And yes, I've been an emergency before, only to be told I was the #7 IFE in progress, and to proceed to a holding fix and hold as published, due to the nature of my emergency being less priority than the other 6 emergencies in progress and airborne at the time.
Having an IFE isn't any fun for the crew involved, but it can be not-so-fun for ATC also, due to their responsibilities. In the heat of the moment, both sides will have to do the best they can with the situation at hand. Hopefully, reading this, pilots can understand what's going on with that voice that may be telling them to "stand by" during their IFE.