In KiCad v5.99 (the unstable development version, nightly builds) it’s possible to import a schematic with standalone Eeschema and a board with standalone Pcbnew. A KiCad project is loosely coupled files with the same base name, e.g. myproj.kicad_pro, myproj.kicad_pcb and myproj.kicad_sch. When you saved the pcb design, did you get the .kicad_pro project file automatically? If yes, you can open that as a project.
Then, when you have the schematic and the board, they should have identical reference designators for the components. If that’s the case, you can use the non-default option in Update PCB from Schematic dialog. In 5.99 it’s “Re-link footprints to schematic symbols based on their reference designators”. Read Update PCB from Schematic's match methods if you want to know details of that process.
Note that I didn’t try importing an Altium project to make sure this works, so if this doesn’t work for you, please tell.