An ATZ, although defined in ICAO docs, has no classification as controlled or uncontrolled therein. Thus the answer depends on the state in which the ATZ is established.
In the UK, an ATZ takes on the class of the airspace within which it is established, and hence it may be controlled or uncontrolled airspace.
UK AIP ENR 1.4 2.7.2
The presence of a tower (ATC) does not, in itself, establish controlled airspace. In the UK, an ATZ may be in
* controlled airspace with ATC (e.g. Stansted EGSS)
* uncontrolled airspace with ATC (e.g. Cambridge EGSC)
* uncontrolled airspace without ATC (FIS or A/G) (e.g. Duxford EGSU)
Controlled airspace is defined as airspace in which an ATC service is provided to, and mandatory for, all IFR flights. Control of air traffic can be exercised in airspace that is not, by that definition, controlled.