No, it's not specifically a British word and, as an expression, 'my mind is all swirly' is very unusual indeed. A 'swirl' is a circular movement, so if you someone says that their mind is 'all swirly' it would suggest that they are confused. As native speakers, we would understand this, but it's not an expression which most of use would use.

As Wayne said, it implies "spinning". But I've never in my life have heard the phrase, "my mind is all swirly". A more common, similar phrase might be "my mind is spinning" or "my mind is swirling".

It sounds like another word for spinning. I would say the person is getting confused by what they are seeing.