What battery class size did you buy? You need a 49/H8 battery class size (the H8 DLG)...if you purchased a smaller class size battery...it may not have enough power. And hopefully they hooked up the battery vent tube.
Also, how long was that battery on the shelf? If it had been sitting on the shelf for a while...it may be slightly discharged...did they charge it before installing? If you've unlocked your cluster...you can go out and do TEST 9 (with ignition in position 1)...and see if the battery's voltage is at least 12 volts...if below that...the battery may need a charge due to it sitting for a while.
The e53 X5 does NOT have an IBS (intelligent battery sensor)...nor the communication bus like the new gen BMWs that require battery registration...don't fall down that rabbit hole.
Some functions do require reinitializtion when battery power is removed...like the sunroof:
The power windows do have a reinitialization procedure...but that is usually needed when like a window regulator is replaced...not the battery...but you can give it a try...especially if you have the correct size battery installed.