Fran could sound like Frahn, but once you put on the umlaut it should, in German, sound more like Fren.
Most of the English-speaking Frances girls I've met use the short a, as in and, for the nickname form, Fran, but the long a (or ah) sound for Frances. Perhaps your mother preferred the nn to the full form, also preferred the ah pronunciation, and made sure that everyone would stop, ask, think about the sound and never, ever get it wrong again. Which is fine in theory, but I do sympathise with all the tedium it causes in your life.
Frances of course means 'Frenchwoman' and has been in use for centuries. But I've never seen an umlaut in Old French. I can see where the confusion could come from, though.