TT23,
If time travel was invented (and I'm assuming that you aren't refering to simple Special Relativity based TT) your question becomes moot.
The sort of time travel that we discuss here goes a lot further than suggesting a failure of causality. Causality violation is assumed. If you can time travel "cause" and "effect" don't have to follow in that order. In fact, "effect" doesn't have to be associated with any "cause" that can be determined.
If you could somehow take the POV of a god-like being who peers into the universe from the outside you could see the date, time, place and by whom the first gadget is invented.
From inside the universe the POV is very different.
For the sake of argument let's assume that the first gadget is invented by you and used for the first time by you today. It is used and improved on for the next 10,000 years. During that time it is guaranteed that the technology will find its way into a period that pre-dates today. Linear time as we normally perceive it becomes irrelevent in the sense that it is necessary to develop and spread the technology.
So "today" comes. Why would you even want to invent time travel? From your POV it already exists and probably in its most advanced form having been transported into your past from some future generation.
Even from some future generation's POV the same problem exists. Time travel technology exists in their past light cone thus they don't have any idea where it came from.
The god-like creature could poll the inhabitants at virtually any time in their history to find a time when they didn't have time machines available...and would be hard pressed to find a time when they didn't have it.
In effect, from inside the universe, it seems to have invented itself.
The upside for all of this, assuming that ones lives in such a universe, is that it wouldn't be paradoxical. During early childhood development when spatial-temporal relationship logic is imprinted on a child's mind these "illogical" causal relationships would be accepted as a normal part of everyday reality.
Because their "logic circuits" would be wired much differently than ours they would probably be able to answer your original question.
The downside relates to a tweak of the Weak Anthropic Principle. In the Weak Anthropic Principle we ask, "Why are we here?" The answer is, "Because the universe has evolved to the exact point to allow us to be able to ask the question."
The tweak is, "Why don't we have time machines?" Answer, "Because we are able to ask the question."