How many days does it take an ammonia spike to occur?

You don't want to wait to see a significant spike before you start dealing with the ammonia that builds up. Ammonia is toxic to fish, and and even relatively low amounts can be harmful to them over time. So you want to test for ammonia, nitrites, and nitrates regularly at first so you know what your tank needs and when.

Cycling a tank is not the same thing as changing the water. The term cycle is a shorthand for your tank's nitrogen cycle, the process where toxic nitrogen-based wastes are broken down into safer ones. Your tank is 'cycled' when the biological filter is capable of breaking down all the ammonia -- the primary nitrogenous waste product -- into nitrites, and then the nitrites into nitrates. There's a great breakdown of the cycle here on the Skeptical Aquarist website.

Your ammonia levels won't be a matter of "safe today, spike tomorrow." Once you add animals to a tank, they're basically producing ammonia constantly. If the tank isn't cycled and doesn't have tons of live plants (which also consume nitrogen wastes), then that ammonia has nowhere to go. How fast it builds up depends on several factors, in particular:

• How much water is in your tank,
• How many animals you have in the tank,
• What kind they are,
• How large the animals are,
• How much they're being fed,
• How much leftover food, fish poop, and other organic waste (including dead fish) is in the tank.

So when people talk about an ammonia spike, what they mean is that their ammonia levels unexpectedly built up faster than their biofilter could break it down.

This is a tricky point in time for your tank: the biofilter needs ammonia and nitrites in order for the denitrifying bacteria to grow, but the fish that produce the waste can't tolerate them. (Many people, myself included, seed a new biofilter with ammonia before actually adding any fish to the tank, but that's a different point.) And unfortunately, it can take quite some time for the biofilter to establish itself -- it took my tank about a month.

Fortunately there are products that will bind the ammonia into a less-toxic form that's still available for the biofilter. I think the best route here is to keep the ammonia and nitrites detoxified this way until the filter establishes itself.

The best one I know of is Seachem's Prime water conditioner: it's nominally a dechlorinator but will bind ammonia and nitrites as well. You'll need to dose daily, but it treats about 10 gallons per mL, so it's fortunately very cost-effective. I don't normally recommend specific products, but Prime is one of a handful of products that are almost universally recommended in the hobby. There are certainly great alternatives that will work as well: the Skeptical Aquarist link above recommends Kordon's AmQuel and Marineland Lab's Bio-Safe.