I read a book that explained this to me. I think things would go on as normal because of the way time works (we think). So lets assume that time is like a book. Each page represents a certain time, You exist in page 300, your grandma and grandfather exist in page 200. You take your time machine from page 300 to page 200, now on page 200 you kill your grandma. and until you go back to your own time you are changing pages. So lets stay you stay from 200 *when you killed your grandma* to page 229, thats when you get back in your time machine and return to page 300. Pages 300-230 will be uneffected and time from 230-300 will be unnefected by the pages you revised (200-229) because pg 230 has already been written and you didn't fuck with it. So only times that you visit you may mess with, the times you don't visit will be uneffected by the ones you do. Because pages that have ALREADY been written can't be changed unless accessed. So to recap, if you visit page 1, you mess with page one. Not page 2. Because the book has already been written. However, what about page 3,000? Does it exhist yet? can you visit it. Gosh the future is the part that's confusing as hell. Not the past.
It's fun to play Dungeons and Dragons as long as you remember it is fantasy. Time as a real dimension is fantasy... Since because ENTROPY.
It goes on about it's own business. The only thing to change in the original timeline would be the absence if who and what went back in time, and all thar would follow such absence (missing person reports and whatnot).