The only way that they won't connect is if your account on the new phone/reset phone is not using the same account on google. If it is the same account, then it should allow transferring between the online account and the phone.
ONLY Do the following if is it is a different account as we don't want you deleting the account with the pictures you want.
If it is a different account, then you would have to cancel the account on your phone (the one without your pictures) and restart the phone. Then when you go into google again, from the phone, use the account information to log on with, for your old account.
Should none of this work, you would still have the option of placing the pictures back on your phone via a computer and a USB connection to the phone, or mailing them to your phone and saving them on the phone.