My MINI looses power at the beginning of second gear until the car warms up a bit. It's not a shifting-too-soon thing, nor does it sound like a "surging" thing.
It's more like I have power, lose it for a second, then get it back. Sound familiar? Is this a fuel injector thing or a plugs thing (I'm due to change them) or what?

 

I say change the spark plugs.

 

Changing the plugs would be a good first step, not to expensive, and easy.

When I changed mine, I noticed power to be more even, and smoother. Originals had 100k miles on them. But I did not have the issue you are talking about.

Mark

 

Another cheap, easy thing to try would be to run some injector cleaner in it the next time you fill it with gas.

Mark