For the time being, turn off Time Machine on your new iMac while you do the following.
Connect that external drive with your Time Machine backup to your iMac. It will mount to the desktop.
In the Backups.backupdb folder, open the folder that bears your Mac Pro machine name. Open the date-stamped folder closest to the day of the fire, and open the Macintosh HD folder. You want to navigate to the following relative locations further down into this drive:
Rename these two folders on your iMac in respective login accounts, and then drag and drop the above folders from their respective Time Machine locations to your new iMac. Test it with just your account first. Assumption: User names and user ID are same on new iMac, otherwise ownership issues will appear as red circles with a white bar on the folders.
Don't forget to turn Time Machine back on and resume backups. You may want to use a new USB3 external drive with that new iMac such as the 1TB or larger Western Digital Passport for Mac WDBGCH0010BSL-NESN model. It is already Mac formatted and Time Machine compatible for $99. Bestbuy (US) has it.