Photos App after Catalina: masters folder is gone, photos missing, new photos named poorly ... what can I do??? Is there another program that I can get to replace Photos?

I used to just copy the latest uploads in the Masters folder to back it up to my cloud storage,

And therein lies the problem, as that accessing of the Masters folder has never supported - there's reason it's inside a package. If you want to back up your originals only to a 3rd party back up service, what you should have been doing is exporting them as unmodified originals, and uploading those. If you do that now the resulting image will have the original filename. Basically, you need to use the app the way its designed to be used.

There are at least 7 or 8 good quality alternatives to Photos. Which to choose would depend on your budget and your exact need. Among the most popular ones are Adobe's Lightroom suite, OnOne PhotoRaw, Mylio, ExposureV, CaptureOne...

I also used to use the Masters folder as well for accessing my images, then placing them into my own folder structure.

I have just managed to export the photos via clicking on My Photo Stream, then

File > Export > Export Unmodified Original (selecting Filename = Use Title & Subfolder Format = None) > Export

then just export Originals.

I will have to import it with Image Capture - upload that to my cloud storage - and then import it into Photos. Does that sound reasonable? 

It does.

If you're interested in changing apps then two in particular spring to mind. One is Adobe's Lightroom - the version with cloud storage. Unlike Apple, their cloud is expressly a back up as well as sharing service, and it has a feature that allows you to download a copy of the originals back to your Mac. So, a workflow that goes from Camera to Lightroom, up to their cloud and this has originals automatically downloaded, and a simple app to back up that download folder to your Cloud storage would automate the whole thing.

Or, more simple, Mylio. It does all of what Photos does, but it doesn't use the Internet. Sharing across devices is done in your own home network, so that there is none of this renaming. That means you can simply upload your newest imports as you prefer - also, there's no package so it's all easily accessible.

If you do decide to evaluate these. - or any app - then grab maybe 2 or 3 thousand photos and use them to test the apps, see if you like them or not.

a folder or catalogue. 

But that's two very distinct things. A folder is just that, an ad hoc grouping files. A catalogue, on the other hand, implies a database that tracks the files. One is simple: Image Capture (in the Applications Folder) will move files from your Phone/Camera/Whatever into any folder you choose, including one you create just for the purpose. But if you want a catalogue, something that offers anything from basic organisation right through to non-destructive processing then that's a whole other things.

If you want just image files in a folder: then you're using the Finder

If you want a catalogue: then explore apps like the ones mentioned above.

If these times are typical for your importing then you have some other issues at play there.

That said, using Photos is an option not a requirement, and you're perfectly free to use another app for the task, as I and many others do.

Bart Hokins is right. It's NOT ok Apple - to do this without warning us. In iPhoto and Photo until v.4 we were able to take the photos to other platforms whiteout using a proprietary program to transform the original photofiles. I don't like it at all. I want my photofiles staying original. At least we have to be warned when this happens.

Really not sure what you're on about here.

Apple has done nothing that affects the user, with or without warning.

we were able to take the photos to other platforms whiteout using a proprietary program to transform the original photofiles

Really haven't a clue what you're referring to. No files are altered at all, and if you're working about the apparent renaming, see my post above.

I truly enjoyed the Photos app until this update to Catalina. Hopefully they will alter things.

I doubt this will happen soon. We are already at the second major Catalina update, macOS 10.15.2.

The new, cryptic filenames seem to be a necessary fix to speed up the iCloud syncing, to introduce new, unique filenames for iCloud Photos Library, that will work across all platforms and system versions, that are using iCloud Photos. User generated filenames have created many problems in the past - ethnic characters or ctrl-characters or characters used by the Unix Shell in the filenames, duplicate filenames, filenames that are too long.