Well, if you are an admin of a Facebook page then you might have tried to upload videos on your page with a mobile phone. And I'm not sure if you have come across this problem or not but I definitely came across this. After finding the solution I thought why not make it easy for other people aswel.
So, as you can see in the picture above that the first picture (mobile version) doesn't have any video uploading option. By clicking on the "Photo" you can only upload picture not videos. But on the right hand side you can see the option of uploading pictures and videos "Photo/Video".
To see your page in desktop version, you need to go on the settings and check the Request desktop site. If you don't know how to do it then you can check here, and make sure your url (at the top) should look like this https://www.facebook.com and not https://m.facebook.com. Because m is for mobile version of Facebook.
This is for android users only and I'm sure about other OS.