One of the meanings of "get on" is to mount a car or to mount a vehicle or to mount etc. But there is not this meaning in the Longman dictionary and Oxford dictionary. Why?
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Dictionaries vary based on whether the lexicographer considers a particular phrase a special case or not. In some dictionaries, they consider "get on" a special case, a phrasal verb. In others, they consider it a normal usage of one of the senses of get that they list.
It's used as readily in British as American English, it's just dictionaries with different approaches.
It's in Cambridge
get on (sth)
phrasal verb with get
present participle getting
past tense got
past participle got or (USA) usually gotten
A2 to go onto a bus, train, aircraft, or boat:
I think we got on the wrong bus.
Pay the driver as you get on.
Show your boarding pass as you get on.
When he heard the news, Simon got on the next train to London.
The old woman had difficulty getting on to the bus.
A lot of people usually get on the train at Cambridge.