Death Ward will trigger the first time you drop to 0 hit points, and Relentless Endurance won't. Then next time you drop to 0, Relentless Endurance will be ready to go.

Death Ward, emphasis mine:

The first time the target would drop to 0 hit points as a result of taking damage, the target instead drops to 1 hit point, and the spell ends.

As soon as you would be reduced to 0 HP, you are instead reduced to 1. As opposed to the Relentless Endurance trait, emphasis mine:

When you are reduced to 0 hit points but not killed outright, you can drop to 1 hit point instead. You can’t use this feature again until you finish a long rest.

You choose when to use it. It doesn't automatically trigger when you are reduced to 0 HP. So when you are reduced to 0, with both Relentless Endurance and Death Ward, just don't use Relentless Endurance. Death Ward will be used, then next time you drop to 0, you can use Relentless Endurance.

There's another point to be made here, though. Even if you want to use Relentless Endurance, while you have Death Ward on, you can't.

The first time the target would drop to 0 hit points as a result of taking damage, the target instead drops to 1 hit point, and the spell ends.

Death Ward triggers when you would drop to 0, and makes you drop to 1 instead. You never actually drop to 0.

When you are reduced to 0 hit points but not killed outright, you can drop to 1 hit point instead. You can’t use this feature again until you finish a long rest.

Since Relentless Endurance is triggered when your hit points are reduced to 0, and Death Ward prevents you from ever reaching 0, you couldn't use Relentless Endurance even if you wanted to. This argument also works if your DM believes you don't get to choose when to use Relentless Endurance, which you asking this question suggests he does.

Yes you can

The rules for wildshape clearly state that you retain any benefits of features from your race if the new form is capable of using them. Since the Relentless Endurance feature does not implicitly or explicitly require any particular body parts (hands, for example) I see no reason why it should not be retained.

The rule on reverting says you automatically revert if you hit 0 hit points. Since your Relentless Endurance allows you to drop to 1 hp instead of dropping to 0, you never actually drop to zero and thus never revert.

... and there is no carry-over damage

The rule on carry-over damage says:

if you revert as a result of dropping to 0 hit points, any excess damage carries over to your normal form

Since you did not revert or drop to 0 hit points, this rule does not apply.