If you are number 10 in the hold, and declare "minimum fuel", you'll still be number 10 in the hold. Here in Ireland the 9 crews ahead of you will not be inconvenienced to accomodate you in this case. You'll be given an EAT ASAP, and informed of any likely changes ASAP. "Fuel Emergency" or "Mayday, Fuel"..... these make you number 1, but your tanks are getting dipped on arrival in all likelyhood.
I'm high level rated, but I know my colleagues in approach are well aware how important getting an accurate EAT to you ASAP is for your own planning. Unfortunately when the airport authority gets snow clearing wrong or when the fire crews have more important things to worry about, that's not always possible.
Where minimum fuel might also help you is enroute, it will make us pick up the phone to the next sector/FIR and ask for something more direct for you.