I'm trying to replace the light bulb in this fixture in my washroom. I was able to unscrew the bulb by slowly moving the bulb anticlockwise squeezing my hand through the glass however I'm unable to screw a new one in the same way. Any ideas on what I should try? Should I try to remove the glass fixture and if so how. I tried to move the glass fixture but it wasn't budging.
It's likely that there's a set screw on the back side of the metal that holds the glass globes in place. Stick your phone up there in selfie mode so you can see the screen while it sees the back side (or you could use an actual mirror...). You'll probably see a little screw to hold each piece of glass. It's possible that the screw is on top instead of around back - that would actually make your life much easier.
Undo the screw with one hand while holding the glass with the other, gently wiggling and pulling the glass until it comes out. Set the glass down, replace the light bulb, then gently replace the globe and tighten the set screw back down. Note that "tighten" means until it's just touching the glass and that the glass won't fall out on its own.
Here's a bit of ASCII art attempting to depict what the metal frame, glass, and set screw will look like:
METAL FRAME GLASS GLOBE
The metal is on the left. The asterisk
* represents the screw going through a hole in the frame, then catching a lip on the glass, which comes in from the right.
There should be some play between the glass and metal so you'll only need to back the screw out part of the way, not completely remove it. Once it's loose enough, you'll be able to angle the glass to get the lip around the screw. Repeat that process on reassembly, and don't tighten the screw enough to break the glass. It's likely that the glass will still be a bit wobbly if you push on it and this is perfectly fine - once you've got the screw just tight enough that it's caught the lip of the glass, gently lower the glass until it's hanging on its own, then leave it alone. The screw will prevent it from falling out and nobody, but nobody, will know it's loose until the next person goes to change the bulb again.