It's highly unlikely to happen. The older they get, the stronger their urge to copy us is! So that, if nothing else, should get LO eating something.
If LO is happy and healthy, a one year old would surely have a good reason for choosing a milk only diet. (I believe so, anyway.)
There's not a lot you can do anyway, if they don't want to eat, you can't force them (I am taking for granted here that you would, in all likelihood, have tried to spoonfeed or help them eat by then anyway)