Right, back again:
Does China still offer two year scholarships to UK graduate students? There used to be a lot, particularly compared to the number of people learning the language. Perhaps something worth looking into.
Re spending a year somewhere:
Kazakhstan is currently the most sensible place for Russian. Huge number of native speakers, including many Kazakhs who speak Russian natively and can not speak Kazakh at all.
For Chinese, should be in China if at all possible. Google areas where Mandarin is spoken natively, and aim for there. I.e not Shanghai or Guangzhou or Fujian (there are of course other areas) as although Mandarin is taught and used as a lingua franca, what people are using in daily life is not Mandarin, and the Mandarin that people are using can be quite non-standard in pronunciation.
I took a few courses in Chinese in China, a couple of months each time, quite some time ago, and the lessons were very rapid and intense. Asbin, definitely worthwhile.
Singapore is also possible, as a pp said, as the usage of Mandarin in daily life has really taken off compared with a few decades ago. That said, the temptation to use English would perhaps be too high.
Taiwan is not such a good idea. China simplified its writing system in the 1950s, whereas Taiwan continues to use unsimplified characters, so one has to put time into learning those as well, which may be useful, or maybe not, depending on one's intentions career wise.
What people who took these languages ended up doing:
Obviously, quite a number got a different job and never used the language again, but this is for those that I know of who did end up using it, and which langage they learnt. Will shorten language name to just its capital letter:
Working in computer science for major tech firms on machine translation and automatic language recognition (R, M)
Engineer - petroleum industry (R, M)
Engineer - industrial manufacturing (M)
Banking (M)
Air steward/ess (M)
Travel industry in one way or another (One set up her own company organizing tours to Lithuania) (R, M)
Interpreting (R, M)
Intelligence services (R, M)
Worth noting: the intelligence services stuff that I knew of were not in the UK (were India and USA) sounded utterly boring. Involved lots of reading newspapers and listening in to communications. Latter was CIA wanting to recruit people studying Russian, they came and gave a recruitment speech to each university class.)