How to physically flatten old prints?

When mounting photographs on matte board I use a hot-mount press. They are a somewhat expensive item, so you could improvise by taking a couple pieces of matte board and a sheet of release paper, and sandwiching the print between them then using a clothes iron. The release paper goes on the face of the photograph and keeps it from sticking to the matte board, and/or some of the print bleeding onto the matte board. Check with the local photo framing and mounting store or supplier or by searching online for hot-mount photo release paper and matte board.

The hot-mount press gets to about 200° if I remember correctly, and pressure is applied for about 60 seconds at that temperature to adhere the print to the matte board with the adhesive paper. (you will not be using an adhesive photo mount paper)

I would start with the iron on a medium setting and gently running it back-and-forth on the matte board so that it never sits in one place but is always moving and applying moderate pressure, covering the entire area of the print. It may take a few runs of trail to figure out how long to iron it, and you may not get them completely flat but you will be able to get the majority of the curl out enough that it can sit in the flatbed scanner. You may also want to take the warm prints and sandwich them between a couple pieces of matte board and wax paper between each of the photos and put some books on it to sit overnight.

You don't need to close the lid. Leave it open and use a large book to press the photo. Don't choose a too heavy one or the glass will bend slightly.

With the lid open the only difference is that empty areas are black an not white.

You can MAYBE soften the prints by leaving them in a very humid (I mean almost saturated) container, but avoid condensation because it would ruin them or make them stick to the scanner glass.

Borrow the glass from several picture frames that are smaller than your scanner. If you are afraid to cut yourself you can put some tape around the edges of the glass or wear gloves. Make sure glass is clean and dry. Lay the glass on top of the the curled photos on the scanner. If you move the glass the photo will move with it so you can get it as straight as possible on the scanner. The glass position does not have to be straight, just the photo under it. Since glass is thin you should be able to close the scanner lid and scan as usual. The glass allows you to see exactly where you position the photo on the scanner. If the weight of the glass is not enough to flatten the photo, then place a heavier object or a second sheet of glass on top of the glass and scan with the lid open. This has worked for me to get good scans, but does nothing permanent to the photo. Don't try this if you think the photo is too stiff and will crack.