What should also be mentioned as a tangent is not just if New Latin becomes a more viable linguistic variety of Latin serving as a lingua franca (hence its survival), but also what happens to the traditional pronunciation of Latin for native Anglophones. This may sound trivial, but it's actually of huge importance - for a long time, Anglo-Latin basically followed the same trajectories as English speech (despite its origins in the Latin spoken in Old French and Old Norman speaking territories), so that as a result thanks to developments in English phonology (above all the Great Vowel Shift) the Latin used in English-speaking countries was difficult to comprehend when spoken for non-Anglophones. The end result is that around the turn of the 20th century (so around the time New Latin goes out of fashion and hence its end as a lingua franca) advocacy grew for a revival of the old classical pronunciation. Now, if the vowels of Middle English were conserved as part of the pronunciation of Latin in England and English-speaking countries (while the consonants continue to go their own merry way), there would be a start in making New Latin comprehensible all across Europe and would be of much benefit in keeping New Latin as the lingua franca.
I'm not sure that the pronunciation would be that important, TBH -- 99% of the communication would be done by writing, anyway, so it wouldn't matter too much if you could understand the other guy's accent. Or if it did matter, then I'd expect the various pronunciations would draw closer together again to facilitate communication.