My doctor is sending me to a specialist regarding a lump under my armpit that may be skin cancer. How fast can it spread if it is not removed soon?

Depend on what it is. First thing first. The answer will depend on what the lump really is? See the specialist and have a biopsy done & other work-up then discuss result with your doctor. A lump under the arm pit can be caused by many things- infection, inflammation, cancer both solid and blood calncer etc. So first to see if this is cancer, if it is, then what cancer? Breast vs skin ?Etc, then what stage? Etc. Etc, .
Tissue Diagnosis. This needs a biopsy to determine if it is even a tumor (benign or malignant). Sometimes other processes can mimick tumors (eg, reactive lymph node, atypical inflammation/infection). A biopsy will tell for certain. Also, in a women with an axillary mass (depending on the clinical appearance) breast imaging and at least a breast exam would be warranted.
Depends. The rate at which melanoma travels varies. We generally feel that weeks do not matter in waiting, however you should not wait months.