Unable to upload image using the Wordpress admin 'Insert Media' option

I am running WordPress 4.3 and in the admin I am no longer able to upload images using the standard upload image tool. I am taking over a project from another developer so don't know exactly what he/she has changed.

The 'Insert Media' box pops up, however when I click the 'Select Files' nothing happens. I would expect it to browse to a folder on my PC to allow an image upload. There are no console log errors and I cannot see the best way to debug this - can anyone suggest my next step to debug this?

via Chebli Mohamed