Below are the things I would check first to see if there is any play.
Ball joints
Unit Bearings
All tie rod ends
track bar bushings
control arm bushings
There is probably something in one of those that is causing the issues. How well balanced are your new wheels/tires? I would assume it's something on your vehicle that's causing it since it's happening both before and after putting the new wheels and tires on.
Just to add a little bit more.
Ball joints/Unit bearings can be checked by jacking up one side and putting hands at 12 and 6 o clock on the tire and pushing/pulling the tire in and out (push top in, pull bottom out at the same time, switch directions, repeat). You may be able to physically see the play or at least feel it. I just did ball joints on an axle and I could physically see the upper ball joint moving and could 100% feel it when wiggling the tire.
The other things to check should be able to be identified with the vehicle on the ground and off; get someone to turn the steering wheel slightly in either direction. You should be able to see movement of a TRE/track bar/bushing. That, or put your hand on each component and see if you can feel any popping or play.
Whatever is loose or worn, that's the thing to start with first.