Hi,
I'm so glad that you posted this message because I have the same problem! I too have struggled to find a solution to it and also find that almost no-one else is tolerant or understanding of it - they just assume that I'm being stupid.
I'm afraid I don't know of any ways to 'jump start' speech. What I have done is found other ways to communicate when my speech refuses to co-operate. I can use the sign alphabet (although that only works if the person with whom you are communicating knows it too.)
Other than that, I either write things down (or preferably type as I am dyspraxic so handwriting can be a problem) or, I take relevant documents and / or objects that are relevant to what I want to communicate and point or gesture (to try to convey, or simulate the action of, what it is that I want).
This generally works well, the only problem is that if you see the same people at a later date and they don't know that you have mutism problems, they generally assume that you were faking it and can be quite unpleasant - I just try to explain this to people and recommend websites, but many people just aren't interested and sometimes, ashamed as I am to admit it, I feel like I just can't be bothered to explain it for the umpteenth time when it is unlikely that there will be a helpful response.
I know that this is silly though because it is only by persevering in trying to make people understand that things will, hopefully, one day get better.
One thing that has helped me is to try and control the triggers - in my case sensory overload and panic attacks. I have mainly done this using medication - which is not ideal, but nothing else worked for me. I have also started taking L-glutathione which seems to have taken the edge of my anxiety a little. I don't know if you would be interested in trying these, but I suggest you do research first, and speak to a sympathetic doctor (if you can find such a person).