I just spent a week in Seoul and many stores there seemed more fluent in Mandarin than in English. This was especially noticeable in touristy areas such as the Myeongdong area and Noryangjin fish market -- in fact, nearly every vendor at the Noryangjin fish market spoke Mandarin (I look Chinese so everybody tries to speak that to me) and it took quite a while to find someone who could understand at least a bit of English. I think this is because of the large number of Chinese tourists that are present in these touristy shopping areas. In general, however, there were more signs in English than in Chinese and a lot of Koreans (especially older ones) have a more positive view of western countries than they do of China.

I imagine it's pretty similar in other countries -- you'll likely find a lot of Mandarin speakers in tourist areas trying to sell stuff to Chinese tourists, but English is likely more prevalent than Mandarin outside of these areas. There are probably exceptions to this -- Singapore is definitely one of them, as a large majority of their population is ethnically Han Chinese.

On a side note -- for the Korea part of your trip, you may want to spend a few days learning Hangul as it is an incredibly simple alphabet to learn. Even though you won't know what most of the words mean, you'll find that a lot of Korean words are very similar sounding to their Mandarin counterparts, and you may even be able to figure out what some Korean signs are saying when there are no English/Chinese translations available. As such, if you're speaking to someone in Korea who doesn't know any English or Mandarin, you might have some luck saying a few nouns in Mandarin and hoping that the Korean counterpart of those characters sound similar.

Singapore

At least in Singapore there is a huge difference between an official language and what people actually speak and understand. Although English is the first official language, many (especially working class) people have a very very limited English level and barely speak a few sentences with terrible pronunciation. This e.g. includes taxi drivers, people working in a hawker center or a supermarket.

Questions like "Excuse me, do you have poultry?" have to be reduced to "Hello. Where chicken?". I was not able to ask for an acid based detergent in a super market because nobody could understand "acid". Often these people only speak their original language (Mandarin or Malay) properly. This is additionally hindered by the strong Singaporean dialect.

Even many educated people have a strong accent, make grammar mistakes often and will fail to understand you, if you speak British or American English. Your conversations will be much more fluent and hassle-free, if you can switch to Mandarin.

Recommendation: If you approach a Chinese looking person, which are the majority of people in Singapore, speak Mandarin.

(Source: I have been living in Singapore for two months)

You would be surprised to visit Malaysia and Singapore. Not only they can speak Mandarin, but other dialects too (Cantonese, Hakka, Hokkien ...etc.).