Yes, both British people and all other native speakers of English use the form have got.
It's very simple.
The verb have meaning to possess or have in one's possession has two forms in the present simple tense.
He has a car.
Question form: Does he have a car? and Has he got a car?
Also, to have to do something (obligation): He has to do the work.
Does he have to do the work? Has he got to do the work?
There is no such thing as using have got in the progressive tense. "have got" is a form of the verb have and it is not a verb (bare infinitive) in its own right.
There is only using the verb have in the progressive tense in terms of have to do something.
The progressive tense is:
He's having to do the work now. [have to do something]
The bare infinitive of have meaning possess does not use a continuous tense with that meaning. (We don't say: "He's having a car today." generally speaking, unless it means something slightly different. "He's having a car today instead of taking the bus." There having a car means someone got him a car with a driver, for example.]
You either have or don't have or haven't got a car. Have is not an action verb.
Naturally, that is NOT: to have tea or lunch where have functions like an action verb.
We're having lunch at noon.
There are other idioms as well: We're having a ball. [meaning:having fun]
Please note: the Brits don't use gotten. Therefore, in British English only, have got can be present perfect or present tense.