Ah, that is harder then!
So, this is a multi-part answer.
- for the long term, the writing system needs to be encoded in Unicode. There probably needs to be some evidence of demand and usage (or desired usage) by a number of people to help that happen. After that, somebody will need to put together a proposal to the Unicode consortium to encode the characters.
- in the short term, you'll need to either use Private Use Area codepoints, or map the letters to Latin. Both approaches have pluses and minuses, but I recommend the PUA approach. If you use PUA codepoints, it will be easier to convert text encoded that way to “real” codepoints later, because there will be no ambiguity. If you have your language's text using “false encoded” Latin, it will have to be checked carefully to make sure that any real English (or other western language) words are in your text are preserved in re-coding.
What is the name of the language, btw? Why do you think it needs a newly invented alphabet, and that such an alphabet will become popular? I am not arguing against it, but I will point out that when there is already a written form of a language, new proposals are rarely successful unless imposed by a central authority.
Good luck,
T