Assuming a nuclear pulse propulsion spaceship would be launched to space with conventional chemical rockets and on escape trajectory from Earth, and it would only then start the nuclear pulse engine,
Using nuclear power, for any reason whatsoever, is looked upon with great suspicion by the existing nations with nuclear capabilities. There are a lot of countries, including the US, that are skeptical of Iran's nuclear program, that they are currently just building reactors, not bombs.
Project Orion included nuclear pulse units that are in practice small nuclear warheads. Also, this combined with orbital capability, would mean a potential for building ICBMs with nuclear warheads.
Even if you circumnavigate the test ban treaty, the international agreements on not spreading nuclear warheads will apply.
However, several countries have not signed the Comprehensive Test Ban Treaty, of them the nuclear powers Pakistan, India, North Korea and Israel. They have instead signed the Limited Test Ban Treaty. On the other hand countries like France and China have signed the Comprehensive Test Ban Treaty, but not the Limited version.
An example of a country not signing any of them is Saudi Arabia.
The Outer Space Treaty (https://en.wikipedia.org/wiki/Outer_Space_Treaty) bars "weapons of mass destruction" from being placed or used in space. As the propulsion capsules of a nuclear-pulse spaceship are basically atomic bombs, I suspect the OST would apply.
One of the countries that hasn't signed the treaty would make things quite difficult. So far as I can tell, it seems likely that a launch of Orion would be acceptable with the Treaty, but any testing relating to the treaty would be forbidden. It's hard to do without testing, so it seems unlikely to be of much help.
Wikipedia states that the test ban treaty is generally accepted to be the end of Project Orion.
The Partial Test Ban Treaty of 1963 is generally acknowledged to have ended the project.