There is no destruction at the end of a Yuga, so yes they can survive. a complete cycle of all the Yugas is called a Mahayuga. 1,000 mahayugas (4,320,000,000 human years) constitute a single day of Brahma - a single kalpa. A day of Brahma starts with the creation or emanation of the universe and terminates with the dissolution back into the Absolute. The higher worlds survive. So religions can survive from Yuga to the next (It doesn't say if they actually will or not), but they cannot survive after a day of Brahma. The current Brahma survives for 100 Brahma years and then there is the grand dissolution where all spheres of beings dissolve into the Absolute. After a Brahma century of re-absorption, there is a new Brahma and a new emanation.
It would appear according to some interpretations that every new emanation has the same events occur again in the same manner and sequence.
Ramakrishna Paramahamsa said that both Buddha and Jesus were incarnations.
These different religions exist and moksha is possible but very difficult to attain through other religions. Swami Vivekananda said "This Moksha-path is only in India and nowhere else. Hence is true the oft-repeated saying that Mukta souls are only here in India and in no other country. But it is equally true that in the future they will be in other countries as well; that is well and good, and a thing of great pleasure to us." (C.W. V. 5, p 446)