How do I recover lost data without backup?

technically speaking, the data is never lost until it's replaced by other data, even if you restore the device to factory settings

as far as I know there are some companies that can restore the data that was stored

on the device but that is in the case of HDD and SSD drives in general ( Macbooks, laptops, etc... )

apple care can help provide the names of certain companies I think in this field, but I am not sure if the same method can be applied in a smart phone !!

If you don't have a backup, there is no where to recover the data from. It's gone.

Log into www.icloud.com from a computer and see if any of the photos have been synced to iCloud.

It can't be recovered if you restore the device to factory settings. Storage on an iPhone is encrypted. When you restore it the encryption key is trashed and a new one is generated. Without the original key the contents of the phone's storage are irretrievable. It's true that the FBI was able to recover data from an iPhone 5, but it cost them over $900,000. And the method their contractor used would not work on a newer phone or newer version of iOS.

The update did not ruin your phone, at least not from your description. Something went wrong with your phone, maybe software, maybe hardware, maybe cosmic rays. It's hard to say what. But, from your description, it happened before you updated. My iPhone 6 once simply restarted itself while I was using it and, when it finally rebooted, it had wiped itself. I restored from the iCloud back up that had been automatically made the night before and went about my life. There is an old saying about data storage: It's not a question of if your hard drive/iPhone/USB stick will fail, it's a question of when.

Most of the recovery software I've seen for iPhones pulls data from the back up on your computer. Now that the phone has been restored, it's unlikely that there is anything recoverable on the phone itself. You could try a company like DriveSavers (they have a good reputation). They will need you to send them your phone. If they can extract any data, it's going to be costly, on the order of thousands of dollars.

For the future, turn on iCloud back up. If your phone is plugged in at night, connected to WiFi and the screen is locked, the phone will automatically back up overnight. I also have redundant systems for backing up photographs. I like Google photos. I also import them to my Mac by connecting the phone to my computer with a cable every couple of months.

why did this update ruined my phone?? It was working perfectly for the last 2 years.

I'm afraid there may never be an answer to that question. All electronic devices are fallible in some way. 2 years sounds about right for a battery to fail due to usage and age. Likes tires and brake pads on a car, they are consumable and will not last forever.

A question for you though. Why, in 2 years, have you never created a backup or synced with iCloud?