Time Machine Photo Restore

You're restoring a Photos Library, right? You can certainly tell Time Machine that you want to put it on an external drive. But be careful--to avoid damaging the Photos Library an external drive must be formatted in either APFS format (better for solid state drives) or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:

Move your Photos library to save space on your Mac - Apple Support

You didn't say what sort of Mac you have, what OS your Macs are running, or if you use iCloud Photos. So there are a couple of other warnings: when you open a Library with a newer OS, the Library has to be upgraded to handle new features. Then this upgrades Library cannot be read by an older OS. The only way to see a Photos Library on different OSs is to use iCloud Photos.

If you use iCloud Photos, then the external drive that's connected to iCloud really needs to be connected full time. There are lots of background operations, and it can cause problems if the Library is disconnected. Many of us solve this problem on our laptops by having only a smaller Library with only our Favorites connected to iCloud.

Once you have the Library on an external drive, if you are using iCloud Photos you can open a new empty Library on your internal drive, set it to the System Drive, and turn on "Optimize Storage." Then the Library will load all the pictures from iCloud. Optimize keeps smaller files locally to save space, and grabs full sized versions from iCloud as needed. This solves the "don't disconnect" problem and the "not enough room" problem simultaneously.

Let us know how it goes, in case I guessed wrong about your situation...

Oh, rather than using Time Machine, the best thing is to just drag the Photos Library to the properly formatted external drive. Then plug the drive into the new machine and double click.