Being interested in Mechanical and Computation Fields, I believe you should focus your Interests on research areas like Automation, Robotics, Electric Vehicle Simulations as well as Industry 4.O which focusses on everything where AI and Machine Learning Methods are applied to Manufacturing Setup. After BE in Mechanical, you may pursue M.Tech from some of the Good IITs or Institutes like IISc Bangalore, either in Mechanical or Interdisciplinary branches like Robotics & Automation, Intelligent Advanced Manufacturing, Engineering Design, etc. Dont totally move into the CSE Sector, there are many opportunities in the Mechanical Engineering Sector which imposes direct relevance to Computational Methods.

As far as GATE is concerned, you may appear for GATE CSE incase you want to pursue M.Tech in CSE, but you may also prepare for GATE ME, as most of these Interdisciplinary Branches accept Mechanical Engineering these days.