IPhone 5c screen broken but device is recognised by my computer

After my iPhone 5c dropped, my screen no longer turned on; moreover, the screen is not cracked.

When I plug my phone into my laptop, my laptop recognises it, however, to allow access to my photos/videos (that I haven't backed up), I need the screen to work. - I need these photos.

After calling Apple Care, they told me to wait some hours and see if my screen turns on, otherwise I have to go in for a factory reset and everything will be lost.

Take it to an independent apple repairer. They should be able to fix the screen and get your data back. Just google iphone repairs and your local city and you should get some repairers

Ok. You haven't asked a question, but if I was to assume one it would be, what to do?

If the display isn't turning on, but still able to recognize touch input, you could try still entering your passcode/word (of course much easier for a passcode) & see if you should be able to access it from there.