Just wondering how long people usually spend abroad for a 12 month European LLM starting September? Can this be done in 10 months?

When must you arrive and when can you depart?

Depends on the LLM programme. Those that end with the writing of a masters thesis do not always have courses in the last part of the programme (ie. time is for writing the thesis). If that is the case, then you can sometimes write your thesis elsewhere and submit through email once finished. Look at the programmes you are interested in and see what the schedule of courses is as that will determine the in-class face-to-face time required. There are many good one-year LLMs in the Netherlands to note. Good luck with your pursuit!

I had a look at quite a few programmes and most seem to have the last two months (sometimes even 3!) reserved for the Master Thesis. Theoretically you can write these abroad but I am not sure how practical that is. You might need access to physical books at your university (unless you can access a good library in your hometown maybe?). Often, a lot of the articles can be found online, find a university that has a VPN for you ;-)