From the American Bar Association's Model Rules of Professional Conduct
PREAMBLE: A LAWYER'S RESPONSIBILITIES
2... As advocate, a lawyer zealously asserts the client's position
under the rules of the adversary system.
As a former prosecutor, I can assure you that if there is a religion in the legal field, it revolves around the sacredness of "winning."
More seriously, "zealously" is a legal term-of-art and not religious. Non-lawyers can have a legal, ethical, or moral duty to be zealous, such as a business executive seeking to maximize the value of a company for the benefit of its shareholders. Some moral systems dictate zealous action in specific situations. The Chinese language has an idiom, jian si bu jiu, which instructs people to help someone in distress.
If I heard the word zealot, I believe that any connotation would be so slight that it would be overwhelmed by the context of the use of the word.
Conclusion: zealot, and only that form of the word, probably has a slight religious connotation but the connotation is insignificant enough to be unimportant and unreliable.