In my opinion, whilst you could do this without a cloud service, it is likely to be a janky, overly nerdy solution, which ultimately means you spend so much time administering it, that it becomes useless.
so options.
1. google photos – download app to phone, download app to PC, point PC app to store copy on NAS. whenever your computer is on and connected to WiFi, it will update the files on the share. (free and easiest solution)
2. the same with Icloud – Download icloud for PC and set it download photos to the share. As above, whenever your PC is on and connected to the internet, it will download your photos and save them to the share.
If you are not prepared to pay for the service you have to manually delete all the photos off your phone after copying existing library to the share and ensure that you delete the photos off your device regularly This deletes them from the icloud storage, but not your local copy, keeping you within your 5gb limit. Downside – you cant view your photos from years ago on your phone, if you are a cheapskate.
3. Once a week plug your phone into your laptop and set it to backup your photos to the share. This completely removes any cloud paranoia you may have.
I am a bit of a tin foil freak and don’t like having all my information in google services (they have my calendar and email) so i keep my photos on icloud, but if i didn’t care about google tracking my every move i would use google photos as it is a really good service.