The second thing to remember is that legislators rarely know much about any issue we will be lobbying. Even if a legislator has some real knowledge about an issue, she or he doesn't live it, so they have a lot to learn from people like you and your coworkers. Only you have real experience and understand the problem from the inside. It is what you know from your on-the-job experience that makes you an effective grassroots lobbyist (that, plus your union power!).
One last point: Never try to "play politics" by making threats. Legislators know that you care about their position, because you've made the effort to express your concern about the issue.
Introduce yourself ...
name, job, workplace, UNAP member, constituent. (For example: "Representative XXXXX, this is Mary Jones. I'm an nurse at XXXXXXXX Hospital, where I'm a member of United Nurses & Allied Professionals. I'm also one of your constituents. If now's a good time, I wanted to take about one minute to ask for your help.")
Express why you are calling ...
the problem and the legislation. (For example: "In my hospital, we see severe staffing shortages on a regular basis, and it's getting worse. We simply are not able to treat our patients with the professional care they deserve. I'm calling to ask your support for bill number 0000, sponsored by Senator XXXXX. (or, if you're speaking with a representative, bill number 0000, sponsored by Representative XXXXXX). This legislation will help us do our jobs by establishing professionally based staffing and acuity standards for all our hospitals. I am calling you as a constituent, and because you're a member of the XXXXXXX Committee (or XXXX in the House) it's being sent to. I hope you can support it.")
Let them know you would be happy to talk about it more to answer questions. If they want more information, let them know you will be in contact with your union to have more information sent.
Thank them for their time and consideration (or commitment).