Photoshop CC 2024 takes ages to open any file and severely limited saving format options.

The images are set to RGB,32Bit color,

Just remember that the PNG description 'RGB 32 bit colour' refers to 8 bits integer /channel for each of RGB + 8 bit Alpha. That is equivalent to Photoshop's 8 bit/channel.

If you set Photoshop to RGB '32 bit' then you are setting 32 bits floating point / channel which is a completely different bit depth and you will not see PNG, bmp or jpeg as a format to save that bit depth.

If I open a JPG file in Photoshop and just hit "save" instead of it just saving over itself, it opens the save as dialogue instead, and, of course, only formats that can't be opened in any other program are available.  This is with making no changes, just opening and hitting "save"

Just need to be absolutely clear about this, because those file formats you list are those that do support 32 bits per channel. The others don't.

32bit color in 8bit colorspace (16.7m colors), not HDR (that's in a 10bit colorspace (1.07b colors).  The same as they were when Photoshop would let me save to PNG.

Is that 32 bits per channel, as in HDR? Don't think PNG supports that.

I start with a multi-layered PSD image.  I make my filters in Filter Forge, then get down to the manual work.  When I am happy with my texture I flatten the image and save it as an uncompressed PNG file.  I have been able to do this successfully, until now.  Photoshop will not let me save it to PNG and it won't let me save it to any file that could be opened by any other program.  This means I will no longer be able to get my work out of Photoshop and into the game.  Also, I can't use most transparency modes anymore.  I used to use screen, multiply, and overlay a lot in my work, and all of these transparency modes are now always greyed out.  The images are set to RGB,32Bit color, like they were when I could save to PNG, and use screen, multiply, and overlay transparencies.

You need to give us some details about these files. What's the pixel size? Layers, transparency, channels? What else do they contain?

I tried "save as copy" and still can't save BMP, JPG, and PNG.  I tried enabling legacy save as copy in the file handling section of the preferences, and still can't choose BMP, JPG, and PNG.  I have tried everything I could, but it's impossible to save to my desired format.  Although I have other programs that can open TIF files, TIF files saved in Photoshop can only be opened in Photoshop.  When I try to open it in any other program, it just says it's not a valid TIF file.

File formats that do not support all properties of the document are under Save A Copy.

Either make sure your file complies with the file format specification, or use Save A Copy (discarding unsupported data).

TIFF is an Adobe specification, so if other applications can't open it they're doing something wrong. 

Taking five minutes to open a file is unheard of. That's a separate and bigger problem, and you may be asked to provide a sample file. Something's definitely wrong there.

I tried saving it as a *.TIF file, to then open it in another program and save it as a *.PNG.  any TIFF file made in Photoshop can ONLY be opened in Photoshop. The same goes for other formats.  No other program will open them, they just say it's not a valid. TIF file.