Yes, you can join B.Tech program after completing your diploma in mechanical engineering. Because diploma is not sufficient there is a lateral entry to pursue engineering.

Lateral entry scheme is intended to admit meritorious diploma holders to the second year of the B.Tech courses to acquire a degree in engineering. The top notch colleges of the country have the provision to Lateral entry for diploma. They have a fixed number of seats allotted for students. Students enter directly through lateral entry, complete their engineering in 3 years /6 semesters. The exam procedure, the course and the degree everything remains the same for the applying candidates.

There is LEET exam for each state and therfore you can apply it once your diploma is over.

Let me know if you want to know for any specific state.