In a nutshell, the answer is yes, most mobile devices have Y2038 issues, and will continue to have such issues, likely up to and including in 2038.
Most mobile devices today are based on 32-bit technology, but there is a major shift underway in the mobile device markets to move to 64-bit technology. Some high-end devices are already using 64-bit microprocessors. Chances are good that the majority of cell phones and tablets will contain 64-bit processing chips by 2038. However, using such chips does not solve all Y2038 issues, for several reasons. First, 64-bit processing chips contain multiple cores, some of which may be 32-bits. As a result, there may be Y2038 issues in the embedded system firmware exchanging dates and times between 64-bit and 32-bit cores.
Another reason why mobile devices may have Y2038 issues in 2038 has to do with the way the fact that mobile devices require precise time synchronization with the current time and may have no easy way for developers to test their applications with future dates.