I did basic searches on ScotlandsPeople for a Frances Dorothy Kemp, just using the surname/ just using Frances Dorothy/ Frances in Glasgow/Lanark/anywhere but no result of any significance. Of course she may have been told she was born Scotland but wasn't. As has been said, if she was 'adopted' prior to 1927 it would have been an informal arrangement so no paperwork will exist. Even later if a family member or friend took her in it it may nor have gone through legal channels. She could also have been illegitimate.
Getting the marriage certificate may give some helpful information, if she told the truth. It may be that if she named her father/adoptive father and his occupation he could be identified on the 1939 register.
