Is there a way to transfer the 360 characters without the 360? Or were the saves hard saves on the Xbox 360 hard drives?
If you are using the backwards compatible version of BL2 on XB1 and had your saves in your Cloud Sync folder, then they should already be available. If your saves weren’t in Cloud Sync but you have backup copies on USB, you’ll need a 360.
If you want to import your saves into the Handsome Collection version of BL2, though, you will need a 360 since you have to use the in-game menus to upload/download them one at a time.
The only alternative would be to find someone you really trust who has a 360 and BL2 , and have them use your xbox account to do them.
But like VH says , you’ll only be able to do it with characters saved in the cloud.