Apart from LSE, UCL and Kings I am considering Cambridge for my LLM in International Law.

I have read Cambridge has an outstanding reputation in this field, correct?

I will be in my final year when I'll apply. I'm just not sure if my credentials will suffice.
I am a current student at The University of Manchester. I did poorly in my first year (62.5% with 3 2.2s) but my second year has been quite good (69,25% with only 2.1s and firsts, including one 80%). One of my constitutional law essays has been published.

I know my first year does not count towards my degree but since I don't have my final year marks yet, they will obviously count in the application process, I'm just not sure to what extent.

Would you reckon that with a strong PS and strong references, I could make it despite not having a First? I do intend to graduate with a good first.

You never know until and unless you have applied. It does you no harm making an application anyway!