My experience is that, for the most part, the bridge tenders are pros that will do what they can to expedite transit, especially if the conditions are extreme. i.e., you are downbound in a strong current, gusty winds, etc. They sometimes are operating with strict scheduling "orders" and can't always operate "on demand" unless you are carrying HAZMAT, a tug pulling 1000' of dredge pipe or are a LNG tanker.
If you get no response after several hails, try calling the local Coast Guard Station via VHF that covers the area, explain who and where you are, what you need, and ask them to ping the bridge via land line and tell them you are standing by on Channel 16/13/9 for the bridge response. That, by itself, sometimes gets you the response from the bridge or they may be able to give you a reason for the lack of response. (tender is greasing the bilge nuts, fetching 100' of shore line, etc.)
That has worked well for me in the past.