That’s a really complex question, since the potential predators and the possible defences vary so much.
At one end of the spectrum the answer it’s instantaneous. This will most likely occur when the indigenous predator is a vertebrate and so has a high learning capacity and a willingness to try new food sources. It will also be high if the invasive species has nor particular defences. So as an example when cactoblastis moths were introduced to Australia and America they were being preyed upon by indigenous bird species almost immediately. They have no particular defence against predators beyond hiding and a slightly unpleasant taste, neither of which stopped hungry birds from figuring out they were edible. In the same way any invasive but edible grass species will be immediately eaten by the first indigenous deer or whatever happens to walk past.
Vertebrates are smart, they experiment and learn and because of that they can learn to deal with novel food source in under one generation.
As you move up the scale you get instances where the predator is a vertebrate but the introduced species has some sort of effective defence. Nit an insect or plant, but the cane toad in Australia is a good example of this situation. The toad is poisonous when eaten whole. For many decades after its introduction it had no predators at all. However by about 40 years after it was introduced several bird and one mammals species had learned that the toxin is concentrated in the skin and that the muscles and some internal organs weren’t toxic enough to cause problems. These predators have taken to skinning or disembowelling the toads, and the behaviour is spreading.
At the top end there are instances where the predators are insects and the invader has chemical defences. Insects don’t learn much. They tend to select food based purely on instinctive programming, and if a food source doesn’t smell or look like food they won’t touch it. That doesn’t mean insects never touch invaders. I’m sure that indigenous ants will happily chow down on cactoblastis caterpillars if they can, but that’s because the will eat anything that moves too slow, it’s not learning or adaptation. In most instances insects will either be pre-adapted to attack an invader and will do so immediately, or they will never learn. Waiting for local beetles to attack a weed is hopeless. An insect species could be left with a food source forever and it will never eat it until a mutation occurs that causes it to try, and such mutations are rare.
That’s further complicated by any possible chemical or physical defences on the plant. Even if a mutation did occur that would prompt local beetles to be attracted to the smell of loosestrife the plant will simply poison any beetle that tries to eat it. For the beetles to have any effect they would need to mutate to deal with the poison first, with absolutely no evolutionary benefit, and then later mutate to be attracted to the smell of the plant. Needless to say that’s pretty unlikely to happen.
Exotic beetles have an advantage because they evolved alongside the plant. They began feeding on its ancestors when it wasn’t poisonous and every time the plant evolves to alter the toxin just a little the beetles just have to evolve a little two. The two species progress together with neither side pulling so far ahead that the other can’t catch up. When the plant becomes an exotic weed the local beetles are so far behind in the race that it would take multiple massive mutations to give them a chance unless the plant is very closely related to an indigenous species that the beetles already feed on.
That’s not the problem, I’m sure that if they aren’t toxic they are eaten by local birds. The problem is that the beetle at home is kept in check by thousands of bacterial diseases, viruses, parasitic worms, insect predators and competitor insects that have aevolved alongside it. Back home the trees have alo evolved to stop the beetle eatng them in avariety of ways. Here it has a smorgasboard of defenceless trees, no diseases, no competitors that can handle it and no insects that eat it. T heonly thing that it has to worry about is a few birds. But bords only lay a few eggs a years, while the beetles lay millions, and th ebirds can’t just breed up by eating bettels because come winter all th birds willstrev to detah while millions of bettle eggs hatch into a wolr dwith very few birds. The birds on their own can’t keep up no matte rhow many beetles they eat, they need othe rpredators and diseases to help them.