My mom is having difficulty getting to the bathroom on time to pee and keeps wetting herself. Any suggestions?

Vinegar in the wash water, I haven’t tried Borax but that would work too.

My mom will not wear pads or Depends either and she’s convinced Depends have latex in them (she’s allergic to latex). I’ve looked at the package of Depends thoroughly and I didn’t read latex anywhere. ~sigh~

It could be the type of briefs you tried. You can try a disposal adult brief made for women such as Prevail. As others mention, you can also try bladder pads or liners that can be used with her own undergarments such as Abena bladder control pads. If she is soiling the bed, underpads can also be an option, these can be on top or under the sheets (recommend on top). These come in disposable as well or reuseable/ machine wash.
As far as the odor, Med-aire makes a product line specifically for incontinence. It is a spray and will eliminate any smell. When cleaning up, use gloves and add vinegar to the wash to help neutralize smell in clothing and in the machine.

I used Hibiclens to remove germs when DH soiled his pads. Just a capful in the washing machine load does the trick.

Other than that, have you tried a bedside commode? Medicare will provide if her physician prescribes the need.