Unfortunately, this sort of bureaucracy is very hard to avoid. You can let the client know you're in a rush as many times as you want, but you will likely be ignored. And the failure of the campaign becoming irrelevant will fall on you.
Something that has worked for me in the past when facing similar situations is to create a timeline of announcements planned for the year. This allows me to limit the waiting time by having announcements planned months in advance, and giving the client enough time to go through the lengthy approval process while still ensuring our news remains relevant.
Of course, sometimes things come up that can't be planned, and in those situations, we have to go with the flow. But truly, the key to successful PR is strategic planning—and a pipeline of news is your best friend!
I would be happy to give you a hand in understanding how to create a successful pipeline of announcements. I'll be happy to take your call!