I am a BS student with a GPA of 4.91/5 and decent research experience. One of my works on neural feedback control have been recently submitted to an international conference. The other one which is more complex and interdisciplinary (CFD + nonlinear dynamics of fluid-structure interactions and control optimization) is expected to yield a peer review journal publication in this year. I had a paid global internship experience at KIMM in South Korea last summer and I am currently holding TA'ship (consultations + HW grading for juniors) at the Applied Mechanics department. My thesis supervisor is quite world-renowned in his field and my final thesis will be related to turbulence modelling and stability of structures with fluid interactions. I am finishing in January 2020 and willing to apply for direct Phd positions. Any suggestions on how and when to contact professors for direct Phd positions are highly appreciated. Thank you in advance!
I would advise starting to research things and interact with any hot choices now. I don't mean asking "will you take me". But figuring out what you want to do, what subfields interest you, geographies, etc. If there are some key groups, you could reach out and ask for preprints or the like. I would not restrict yourself to people that your current advisor knows, but certainly leverage it.
If you were in the US, I would even do some informal travel (on own dime, but with notification). However, I would not bother from overseas. But do start to educate yourself at least and/or start to build some bridges for key areas.
In the US, almost everything runs with starts in AUG/SEP. It is unlikely that a place will start you in a cohort that starts halfway through the academic year. That said, I have known students that got accepted and then got temporary positions doing research or the like for a half year.
Given 2, in all likelihood, you would be starting ~AUG 2020, with an application due ~31DEC19 (with submission possible in second half of 2019). You can investigate/request applications now. Some schools may already have them out, some may wait until June.
For what it's worth, you may be able to (mildly) sell your early completion as an opportunity. I have seen research groups take people like that (hoping, but not requiring that the person picks their group during advisor selection). You can also use the time for personal things (family, work, independent study, hiking). But if at all possible, I recommend to use some of that "free time" (the summer at least) for work/internship at the school you are heading to. Will give you a small leg up on the process. And also will help with English improvement and cultural assimilation.