Note: What follows is in absolutely no way practical or useful. This is a Rube Goldberg approach to rocket propulsion, starting with conventional rocket exhaust and ending with nuclear fission.

This won't work.

With that in mind (that it won't work) let's see where it goes...

Fission

The simplest, easiest, and most common way to induce fission in a heavy nucleus is to add a slow neutron, and those don't really occur in rocket exhaust. Neutrons are uncharged and so they can just wander quietly into the nucleus and make it unstable. Charged particles like ionized hydrogen in rocket exhaust can not even get close to a fissionable nucleus because of the coulomb barrier, and protons in general move heavy, neutron-rich nuclei closer to stability, not farther from it. It can possibly happen but really proton-induced fission requires very very high energy, nothing like thermal energy.

However only very high energy hydrogen isotopes can be made to undergo fusion, and that usually makes neutrons, and that could induce fission.

Nothing here is even close to practical. It's so far from practical I won't spend time thinking about it.

Fusion

You could imagine using the highest energy hydrogens to undergo fusion in an absurdly hot rocket exhaust. Energy of only say tens of keV to a few hundred keV are considered for fusion reactors.

That's like 100 million Kelvin to a billion Kelvin. Not a likely temperature from a traditional rocket exhaust. But if it happened, it would make thermal energy. As a side benefit some isotope combinations would provide neutrons which could then induce your fission. But fusion makes fast neutrons, so you'll need a neutron moderator in your nozzle as well.

You could use the thermal energy in traditional thrust to generate electricity, then use that to accelerate hydrogen isotopes to a few hundred keV as they do in neutron generator tubes Those neutrons could then be slowed with a moderator, and used to produce your fission.

None of these are remotely practical.

Slightly related: How did Northrop Grumman propose to make the Global Hawk nuclear powered?

Source

Note: What follows is in absolutely no way practical or useful. This is a Rube Goldberg approach to rocket propulsion, starting with conventional rocket exhaust and ending with nuclear fission.

This won't work.

With that in mind (that it won't work) let's see where it goes...

Fission

The simplest, easiest, and most common way to induce fission in a heavy nucleus is to add a slow neutron, and those don't really occur in rocket exhaust. Neutrons are uncharged and so they can just wander quietly into the nucleus and make it unstable. Charged particles like ionized hydrogen in rocket exhaust can not even get close to a fissionable nucleus because of the coulomb barrier, and protons in general move heavy, neutron-rich nuclei closer to stability, not farther from it. It can possibly happen but really proton-induced fission requires very very high energy, nothing like thermal energy.

You could think about using fusion power from the hydrogen isotopes in exhaust. However that still needs much higher energy than you normally find in exhaust.

The fusion usually makes neutrons, and that could induce fission. But nothing here is even close to practical. It's so far from practical I won't spend time thinking about it.

Fusion

You could imagine using the highest energy hydrogens to undergo fusion in an absurdly hot rocket exhaust. Energy of only say tens of keV to a few hundred keV are considered for fusion reactors.

That's like 100 million Kelvin to a billion Kelvin. Not a likely temperature from a traditional rocket exhaust. But if it happened, it would make thermal energy. As a side benefit some isotope combinations would provide neutrons which could then induce your fission. But fusion makes fast neutrons, so you'll need a neutron moderator in your nozzle as well.

You could use the thermal energy in traditional thrust to generate electricity, then use that to accelerate hydrogen isotopes to a few hundred keV as they do in neutron generator tubes Those neutrons could then be slowed with a moderator, and used to produce your fission.

None of these are remotely practical.

So like I said...

This won't work.

Slightly related: How did Northrop Grumman propose to make the Global Hawk nuclear powered?

Source