That looks like manual vector node editing to me - in the Affinity tools, you could start with a new Affinity Designer file, set up your typography minus the warp, alt / option-drag to make a copy of that element (trust me here) and then close the file - open the same file in Affinity Photo - apply a Mesh Warp and tweak till you're happy with it (this will get you your forms fluidly and quickly, but will rasterise those elements - which is why we copied them) then save.
Jump back to Affinity Designer, re-open that file, and now you can copy a new version of your text art, select it and choose "create outlines" to convert the text into vector curves, and now you're simply manipulating vector nodes to match your earlier Mesh Warp results and get the final look you want in nice clean vectors.
Affinity has said that bringing the Mesh Warp and Perspective Warp tools into Affinity Designer is on their development roadmap, but it's not there yet - so though it works to just open the Designer file in Photo and warp away, you get a rasterised layer out of it.
Bear in mind that if your output need is just raster pixel art - you need not come back to Designer and vector edit - if this is to create a texture map for a 3D asset for example, you can just save a high-res .png and you're good-to-go. If however you're wanting vector output, for printing, vinyl-cutting, .svg or .eps files, you'll need to move some points around.
Hope that helps.
