If you don't mind me asking, what is the tread timber? and what is the riser finish to be? is it interior or exterior, and what level and type of foot traffic will it have to handle? An interior office stairwell will be different to say a set of warehouse exterior loading bay steps etc. I would suggest you try this...
TEST your chosen timber by fixing a section of it (solid bed method) to a concrete surface using Ardex X77s rapid setting single part flexible tile adhesive with Ardex P4 primer to all mating surfaces. You may need to mechanically clean the step surfaces if they are not new or are contaminated in any way. This stuff dries in under an hour and then you can fix the tread down as you prefer ( you will be surprised at the strength of this bond but don't be tempted to rely on it solely).
Oak dose not like cement so be sure to prime it thoroughly prior to testing it. This primer will seal the mating surface of the timber against any moisture penetration and likely staining from the cement based adhesive components.
I have used this method for donkeys years inside and out and I can tell you it is a bomb proof and very reliable way to fix timber to brick or concrete. You will have zero voids and a perfect very strong initial bond between the cement based surface and whatever timber you use especially in exterior applications. This adhesive is also has pretty good tanking like quality which is also a benefit.
But the thing to do is prove all this this to yourself with a test.
Within two hours you will learn more about sticking your wood to cementitious surfaces than you ever thought you needed to know.
The learning curve will cost you about £40.00 but the knowledge gives you a priceless new understanding for many other applications for setting timbers for all sorts of leveling and straightening applications.
We stick floor tiles down with this stuff, straight onto ply etc... think about what is has to achieve to do that.
Sounds a bit rude but I would call it bedding and screwing
