To become a professor for BSc Computer Science

The course offered by Annamalai University is having approval from UGC and you are eligible to appear NET examination if you meet all other criteria. So you need not worry about that.
But UGC is planning to make a PhD as a must for teaching jobs to graduated and postgraduates. So after completing your PG you may have to go for PhD also.
What I suggest you is after completing your M.Sc, join as a lecturer in any private college. These colleges may not insist on NET qualification. So while working there you can register for your PhD and can complete that course. Then you don't have any problem to become a Professor or Associate Professor or Assistant professor in any college or University. So please think in these lines.

In relation to your query regarding your eligibility of UGC NET test because of your perusal of M.Sc in Computer Science through the distance education, I would like to clarify the following points as indicated below -
1) Annamalai University is recognised through UGC and hence it is not likely to affect your eligibility provided you secure a minimum of 55 percent marks as your final aggregate. This holds good for you being a general category candidate. In case of other categories a relaxation of 5 percent is admissible for the eligibility of test.
2) Your final selection for the post of lecturership to coach the students taking computer in B.Sc stage would depend upon your final scorings in the UGC NET test. Your better performance in the test will lead to your success. Moreover, you have to clear the oral before the board of interview. They would assess your potentials in your related area apart from your general intelligence and your flow of expression in representing the facts.
3) You may undertake a session of enhancing your speaking capability through a reputed institution in the similar line.
4) For a stable career as a professor, you will have to complete your doctorate degree in the Computer Science without making any delay while you undertake your teaching assignment as a capacity of a professor.

You can apply for UGC NET exam and there is not any problem as Annamalai University is UGC recognised. Provides you have Minimum 55% marks to be eligible for appearing in UGC NET exams.

Since you want to become a professor, nobody can stop you from achieving that target bit proper destination, hard work and indomitable will are necessary to herald this success.

For a stable career, little attention towards communication skills is vital to groom your communicative speaking and improve the chances of your success.

Annamalai University is approved under UGC and its distance education courses like M.Sc. in Computer Science are approved by the Distance Education Council, New Delhi. There is one letter presented in the Annamalai University site from Department of Education, Govt of India, where the Department of Education has endorsed its courses as recognised one. So, there is no worry on that aspect and you can go ahead with your course.
Subsequently, you can go for NET exam which is the gateway to apply for lectureship in any higher educational institute. If UGC incorporates PhD as an essential eligibility condition for NET exam then you will have to complete that also before going for that.

As you are currently pursuing MSc Computer Science (lateral entry) through Annamalai university distance education and want to know if this course is UGC recognized, I want to tell you that Annamalai University has recognition from UGC for its regular courses for the current session but for distance mode and open course there is no recognition from UGC for 2020 onwards. For this, you can also check the status of recognition of Higher Educational Institutions (HEIs) for Distance education on the website of UGC-DEB at https://www.ugc.ac.in/deb/.

Distance Education Bureau, UGC has not rolled out the list of HEIs and their courses for new recognition status starting from July 2020 onwards yet. The earlier recognition status was applicable up to June 2019 only. If you have taken admission in the lateral entry course before July 2020 i.e. in January 2020 cycle, then your course is UGC approved otherwise, you will have to wait till the UGC releases the new list of recognized HEIs and their courses.

NET is a compulsory requirement for becoming an assistant professor in any recognised university. As mentioned above, now you are pursuing M.SC course through Annamalai university. And this university is UGC approved and your M.SC in a computer science course is also approved by distance education council. So, there would not be a problem in this teaching field. After completing M.SC, you can apply for UGC-NET. You must have minimum 55% marks in post-graduation for appearing in UGC-NET exam. Moreover, you can enrol for a PhD. A doctorate is also compulsory for being a professor. Lecturership job depends on the performance in an interview as well as scored marks in UGC-NET test.