Hi, I'm not sure if the SAI would make much difference but I was wondering if you had it fitted but didn't have it connected correctly after the rebuild if it would cause cold start problems.
When i first did my vanos rebuild, when i started the car it ran really rough and i had not tensioned the aux belt correctly, after i tensioned it correctly and after a few starts it ran really smooth on idle, i am not sure how the aux belt could give poor engine idle so the two may not be connected.
I wondered if by changing the vanos seals it makes the vanos really stiff to move and if that affected the timing until it warmed up as the vanos does take a while to bed in, did they actually fit new vanos seals ?
the other thing i would check is how well the throttle body is fitted and also if there are any other vacuum leaks.
Can you plug it in to any diagnostics