As far as recovering them off the phone, that isn't going to happen. You could sent it off to a forensic data service... they will charge a ton and might recover some of the pictures...
But in all liklihood, they are all gone.
In the future, you really should utilize a cloud backup service. Google Photos is free, stores unlimited photos/videos under 16MP and 1080P.... and once set up, it's completely seamless to the user.