Hi there!
I have tried to achieve the same effect as in the PSX games, using only unity though with somewhat success. The technique ive used should apply to Unreal if you are able to change the rendering order or Z-depth for your objects: 1. characters and npc’s in front 2. static backgrounds
In Unity shaders are used to change the z-depth so that requires to do some script work, if Unreal has a way to script shaders or change them then that could work also.
Here is a thread about that: Rendering Order Revisited - Questions & Answers - Unity Discussions
Would be cool to see more psx style games!