The allotment of all colleges will start together. For example many students will have opted for a particular college as their first choice. But only the ones with the highest ranks will get a seat in them. Now the highest ranks have been allotted seats so for example in their second choice they would be excluded from that college as they got a seat in their college f first preference. So according to your arranged order of preference, at whichever college a seat is empty at your rank it would be allotted to you.
Counselling as in allotment of seat in a particular college will be intimated online but you have to visit the college to submit documents, fees and take admission. And if you want to remain in second round for a better college you have to inform the authorities in the first college while taking admission.
Seat allotment in private colleges will also take place on the basis of NEET ranks. I think they have independent procedures to apply for but the final list should be based on NEET performance only.