I have made the assumption (incorrectly?) that from what you've stated the staircase is to be of the closed riser type
By setting out and geometry I actually make your angle 49.2° (based on an overall rise of 2700mm and overall going of 2300mm) which really is a very steep staircase (and too steep to meet Building Regs on a new build in this country, the UK). Taking your requirement to have a 32mm thick tread (which is adequate for a staircase with closed risers) I make the individual rise to be 157mm (bottom step to floor will be 158mm) and the individual going on each tread to be about 135mm - far too shallow for safe traverse. I've made a few sets of replacements stairs for cottages with a rise and going square at about 200mm which makes for better climbing. For your staircase might I suggest keeping a minimum going of 200mm to give enough for a reasonable amount of foot to fall upon. That gives a rise of approximately 232mm and works out at about 12.6 steps...... so rounding off gives 12 steps with a rise of which gives you a rise of 207mm per step and a going of 178.5mm, which if you add 20mm for a nosing and overhang makes for a tread of 198.5mm depth. The bottom step would have a rise of 208mm in this scheme and the overall length of the stringer works out at about 3500mm. If you have access to CAD software you can confirm these figures or I'd be happy to publish a drawing of the going and rise so you can make-up a template for setting out.
Edit: For those in the UK or Ireland; BS/IS (British Standards/Irish Standards - the regulatory bodies setting the code on these matters) regs stipulate a maximum angle of 42° on domestic staircases (new build) with a minimum going (step depth) of 250mm and a maximum rise per step of 190mm. The general rule of thumb is that the normal relationship between the dimensions of the rise and going can be expressed as detailed in Approved Document K, which states that twice the rise plus the going (2R+G) should be between 550mm and 700mm. In this case we have a figure of 2*207 + 232 = 649mm
The risers will only need to be 6mm or 8mm plywood and should be grooved into the underside of the treads and held in place by wedges. Similarly the treads should be glued and wedged into place. Properly built with sash or pipe cramps there is no need for screws, which in any case are potentially a source of squeaking as the stairs get older.
BTW 15mm is deep enough for the housings (grooves), although I normally just go 12mm, but a stronger joint will be made by using a stair housing cutter such as these
which are actually a shallow dovetail
If these steps are the open riser type then the setting out rules remain the same and I'd still suggest retaining the same going, but the housings for the treads cannot be secured by wedges and a straight cutter is used. For maximum strength a partial riser beneath the nosing of each tread is still required in our Building Regs and this needs to be set into a housing on the underside of the treads. Once again I'd be happy to supply a drawing. Personally I wouldn't use screws, but instead I might let-in two or three stainless steel tie rods hidden behind the part risers with threaded ends and domed nuts at each end.
For those not in the know the term "housing" refers to a groove or dado in American terms and is the Anglo-Saxon for your term.