So I assume you have not been using iCloud to sync your photos then (or they would still be in your online account)
However they should still be safe on your old mac, but haven't yet been transferred to your new one.
Since you still have your old mac.
1 - Start it up
2 - open photos on your old mac (are your old photos still all there?) and go to preferences. That will tell you where your photos library is located as show here (circled):

3 - now can you find that library using finder. Check the size, so you know how much space you need to copy it.
4 - copy it to an external hard drive using finder (the external drive should have enough free space, and be formatted APFS or MacOS extended (journaled)
4 - Plug the external drive into your new mac
5 - Copy the library file into your new mac pictures folder. If the library you already have there has the same name as the one on your old mac, rename it first.
6 - Open the library you have just copied to your new mac (Hold down the option key while starting photos and select it from the list - see sedond screenshot). You should now have all your old photos on the new mac.
If that has worked - NOW - SET UP AN AUTOMATIC BACKUP TO MAKE SURE YOU DON'T HAVE THIS WORRY AGAIN. See:
https://support.apple.com/en-gb/HT201250
