I've had my x28i M Sport since yesterday. I'm not sure folks are going to like my report. The X1 does not have tip-in issue reported on the 2011 5-series, IMO.
The throttle action is no different than our 2009 128i, 2009 335i (sold), or my parent's 2012 328i (drove it today). This is just how BMWs are these days.
I've been driving BMW's since the late 90's (E36, E39, E46). There was a difference in the way the throttle responded to the pedal inputs between those older 3 series and the E39 5 series (by my recollection). The 3s were more responsive and sporty than the 540i which had more power but throttle responsiveness was smoother and less sporty.
When someone ran a red light and totaled my E46 I purchased the 2009 LCI E90 335i (my first BMW drive by wire). The throttle response reminded me of the old E39 5 series, tuned a bit more to feel luxurious than sporty. I felt like the E90 was an amalgamation of the E46 and E39. When my wife's 2007 Honda Accord was totaled, she got the 2009 128i coupe.
In BMW D/Drive there is some minor throttle delay (I estimate no more than 1/4 sec) but no more than in the Accord or in many other rental cars I've driven (most recently a 2012 Maxima). Again, this slight delay makes normal driving (cruising) more luxurious/smoother. In DS/Sport the throttle response improves greatly and is quite sporting.
I have a 2001 Honda S2000 with Greddy Turbo (very little lag) that I has been my daily driver for the last 8 weeks. This vehicle is the furthest thing from drive by wire you can find these days.
I am convinced the perceived X1 throttle delay is due to the 8AT transmission tuning. In D/Drive, the transmission is in such a high gear and must downshift when you depress the throttle. Taking off from a light, the car idles and takes off in 2nd by default. Feathering the gas takes off smoothly. Mashing the gas in D/Drive has a delay because it has to downshift and rev match. In DS/Sport, you can take off from a light with no issues. Same things happen during normal driving. The transmission in D/Drive cruises in such a high gear that it must downshift when you mash the throttle, causing additional delay.
When you are in DS/Sport, the transmission is already in an appropriate gear and often doesn't need to downshift when you give it some gas. We don't even need to discuss ECO PRO. Don't use it unless you are cruising on the freeway, it completely mutes responsiveness.
So far, I am very happy with my decision to trade the 335i for the x28i. The 8AT transmission is awesome so far. I was able to use the paddles to click from 8 directly into 4th...impossible to do with the 6AT.
Sorry for the long post. I hope it was all coherent.