An air carrier Flight Crew declared an emergency and diverted to their alternate with barely adequate fuel following their failure to obtain expedited handling at their destination airport despite having declared 'minimum fuel'.

Narrative

We arrived in the ZZZ area with some holding fuel and weather/runway conditions adequate for landing; then received extensive vectoring for XXR. Runway conditions deteriorated making landing attempt that runway unavailable; declared minimum fuel and requested Runway XXR; which was active and above minimums based on reported weather with braking action good. We restated minimum fuel but still received lengthy vectors for XXR. While on final XXR two preceding aircraft executed missed approaches due to weather although ATIS was still reporting well above minimums. We finally declared an emergency and requested immediate divert to to our planned alternate which had VMC weather and no precipitation. Planned on arriving there with 4000 lbs but due to greater than planned burn during second approach and enroute to the alternate we landed with about 3000. At no time while maneuvering for the two approaches at ZZZ did ATIS or ATC provide any information that weather had deteriorated. Declaring minimum fuel seemed to have no effect on our sequencing; even after declaring an emergency for fuel we had to request more direct routings to alternate. Accurate weather info during rapidly changing conditions would have resulted in a much earlier divert and improved safety margin.