Can a person become a professor in English without having a Bachelor degree in English?
Of course you can.  Yes, once you complete a Ph.D. in English, you will not need a bachelor's degree and you will not need a master's degree when you search for a job.

Entry into a Ph.D. program in English may be done by those with bachelor's degrees in other, related fields.  Or by those with appropriate experience outside academia.

And let's face it:  Once you are awarded the Nobel Prize in Literature, practically any English department will hire you as a professor, even if you have no degrees at all.