Alright, sorry for the massive bump but I felt that I should update all of you on this topic as I have now upgraded.
This should serve as the solution (although I can't pick it as the solution, so I'll be picking the closest) since I have tested the game out with my upgraded PC.
So first of all, to clear things up, the answer to this question is no.
My initial PC setup was not able to run Battlefield 1 even on the lowest of settings.
So, what I did was I looked into upgrading, and after a little over a month, I have finally done so.
The graphics card I went with is the GTX 1060 3GB, and along with that I added another 8GB of RAM (making my setup 16GB, running in dual-channel).
Now to the performance aspect;
I haven't benchmarked any exact frame rates as I was on vertical-sync for the most part.
But to put things into perspective, I was on Ultra for all of the rounds that I played. I was able to play on several maps, including Amiens, Ballroom Blitz, Sinai Desert and more.
Whilst on vertical-sync, about 95% of the time the game was running smoothly between 58fps to the 60fps cap. The only times the frame rates dropped were on very intensive occasions, such as when a behemoth was dying or when mortars were spammed, and even then the drops only went down to around 50fps - 56fps. Not to mention that these drops are most likely caused by my CPU. There were almost no drops to 40fps whatsoever.
Overall, a very enjoyable experience, and very playable at Ultra 1080p.
If you're in a similar situation as myself, upgrading to a card such as the RX480 or the GTX 1060 3GB can help a lot. RAM is also very crucial in some situations.
Thank you all for you replies. I hope this answers anyone's question(s) in the future.
Cheers!