Depending on the flavor, Americans might call them Fudgesicles or popsicles or ice-cream bars.
American children would not use any variation of I strongly demand a ...
These would be common, from more polite to less polite:
Polite:
- May I have a popsicle.
- Would you get me a popsicle.
Neutral
- I want a popsicle.
- I really want a popsicle.
Neutral, but so, so annoying:
- I neeeeeeed a popsicle.
[Thanks, Chronocidal]
Impolite
- Get me a popsicle.
....and, of course, adding "please" onto any of these would make them more polite, while adding now would do the opposite (unless there is contextual reason for it: e.g. I have finished my homework. May I have a popsicle now?.)