What's the common name in usage in the UK for the part in the hospital that specializes in one field only (for example neurology)? I saw two different names 'wing' and 'department'. But both are in non native English speakers country, so I'm not sure what it reflexes and if it's correct also in context of medicine.
Wing and department are both acceptable.
"Wing" suggests a physical building that is next to the main part of the hosptial. Department suggests part of the organisational structure.
At my local hospital the buildings are named "Jubilee Centre", "Canadian Wing", in which there are multiple "Departments", each has its own name. The Canadian wing houses the "Ross Tilley Ward", "Day surgery" and the "Diet and Nutrition Department". It is possible for a building to house just one department. There is the "Burns and Reconstruction Department", which is both a building and a department.
There are many other words that could be used, and sometimes no word at all. One of the departments is called "Outpatients" another is called "Cardiology".
A specialist section in a hospital is sometimes referred to as a unit as in "the maternity unit" or "the burns unit". Taking the definition from Collins:
any group or individual, esp when regarded as a basic element of a larger whole
So the maternity unit is the group within the hospital that deals specifically with maternity matters. You can see from the ngrams graph that "maternity ward" is more popular than "maternity unit", but that is more popular than "maternity department" (the order still holds if you switch the search to British English).
That said, it is also normal to simply omit the specifier (ward/department/unit) and say something like "Send these results through to cardiology".
If you attend hospital for a specific reason, you are very likely attending a "clinic". A clinic can be a specific building/location but it can also be a specific session. That is, a "baby clinic" can be held in a rural community centre, or it can be a specific part of a hospital building.
Have you considered the word "ward"? That's how I've heard it mostly commonly referred to.
I am in the UK, if that helps.
A wing is a large section of a building. It may contain multiple departments or wards but it's a term common to all buildings large enough to have them. It's not hospital specific.
A ward is a room where patients sleep in a hospital, it may or may not be bound to a specific department. A specialist ward may be, but many wards are general and will take patients from multiple departments
A department is probably the word you're looking for. This covers the likes of cardiology, physiotherapy, ophthalmology etc. They may have the word department in the name, but will often just be referred to by their 'ology' or sometimes as a unit.
Generally 'department' relates to an administrative division which isn't necessarily located in one specific place while 'wing' tends to refer to a large area of a building or complex of buildings.
For example you might say that the cardiology department is based in the east wing of the hospital.
Of course if one part of the building is dedicated to one department it might well be called the cardiology (or whatever) wing.
There is also 'ward' which refers specifically to a (usually large) room or suite of rooms where patients are housed and/or treated.
You often also see the term 'unit' which tends to be used for a composite collection of staff, equipment, space and other resources intended to work together for a specific function or specialism. The implication often being that it is able to draw on a wide range of resources and coordinate them effectively.