Take RL Earth's atmosphere and leave it where it is; then tilt the solid parts of the planet under it.
In other words, move all your landmasses and oceans from where they are in real life to where they are in your alternate world; then just paint real-life's climate over it without paying attention to the land underneath.
That'll give you a start.
You'll have to make some modifications; the first few will be major but obvious, the rest will be less obvious but also minor.
BTW are you planning to leave your axial tilt the same -- 23° 26′? (I like to use 22.5 degrees, "almost the same".)
Is the nutation going to be 100' (1 & 2/3 degrees) as it is *here*? And how often will the planet "nod"? About once every 18.6 years, as it is *here* IRL? Will it match up with the precession of the Moon's orbital nodes, as it does *here* IRL? If the answer to both is "yes", then will the moon, in general, behave as it does *here* IRL in OurTimeLine?
Is precession-of-the-equinoxes still going to be about once every 26,000 years?
If all of these are true, then odds are the seasons and Ice Ages will be similar *there* in the AlternateTimeLine to what they are *here* in OurTimeLine.
Or so I guess; some other responder may know more.
Have you looked at Geoff Eddy's Climate CookBook ?