- British English and American English
It's easy to see why you may be confused. When some dictionaries
mark a definition or an entry with British, it may mean British
Your linked Cambridge dictionary page says "British English", but it does not say or mean that it is not American English. If you look on the top right area of the page, you'll see a globe symbol, and from there you can choose American English, which will show an entry.
However, that's not necessary. To see how Cambridge marks words that are British only, look up petrol in the same way you looked up build up and you'll see UK (US gas) to the right of the pronunciation information.
If you check Cambridge Dictionaries' general dictionary site (http://dictionary.cambridge.org/dictionary/english/), it will tell you to search for your term and then click "British, American, or Business". If you click "American" for the British term petrol, you'll see Br to the right of the pronunciation information. That is how you would tell that a term is British only.
Your link to the MacMillan dictionary has another link on it (which is hard to see): View American English definition of build up to. This shows the same definition labelled American English.
http://www.thefreedictionary.com/ is another good resource for idioms, and lists "build up to". We can note that there is no marker on that result indicating that it's British English. Removing all doubt, we can see that it cites McGraw-Hill Dictionary of American Idioms and Phrasal Verbs.
You can answer this type of question for yourself in the future by checking a few more dictionaries. One good American English dictionary is https://ahdictionary.com/ which has entries for build up and build-up.
- To build up to defined
To build up to (something) is listed as an idiom at The Free Dictionary:
- [for a person] to lead up to something or advance to doing or saying something. I can tell you are building up to something. What is
- [for a situation] to develop into something. The argument is building up to something unpleasant.
Depending on the browsers and search engines available to you, and their settings, you may be able to find idiom definitions by searching for terms such as "define build up to" or "idiom to build up to".
(NOTE: This question is arguably off-topic, but the OP has engaged in a reasonable effort to research the answer and I believe responses may be of use to other ELLs. See the highest-rated answer at Policy for questions that are entirely answerable with a dictionary for insight into my rationale.)