In American English we tend to say cell or cell phone. I'll call you on/from my cell. In British English I believe they tend to say mobile instead of cell.

Yeah, Britain uses mobile phone (or mobile for short), America says cell phone (cell for short). You could say "I'll call you on your mobile" or "I'll call you by mobile".

What you wrote is not correct. You need to say either "I will call you on my/your mobile phone" or "I will call you by mobile phone." As Susan says, we say "cell" not mobile in the US.