If I have the theoretical knowledge required for an FAA ATP, could I use that knowledge to pass the EASA ATPL exams?
If you are asking about knowledge - you can use some points, but generally FAA ATP knowledge is insufficient. EASA has 14 subjects, and requires much stronger theoretical background. Just go through the EASA learning objectives and you will see it.
Moreover, there are some differences in Air Law (EASA exam pretends to be ICAO-based only, but in fact it asks about European regulations as well), IFR and VFR communications, and even units of measurement (for example, EASA has some questions where you pick altitudes in metres, not in feet).
And, finally, you cannot even sit the exams without taking a theoretical course in some EASA-approved flight school. That means that you usually need a course completion certificate before going to CAA. There is another way if you have 1500 hours under your belt, but that is a completely different story, and in that case sometimes you don't need to sit the exams at all.