TDateTimePicker is perfect.
Some more issues related (just to document them, maybe you or somebody else wants to address them as well , if not I can write bugreports, of course...):
TDateEdit's calender is strangely offset to the left if the edit is too close at the screen right. Moveover, if the edit is near the screen bottom edge the popup calendar overlaps with the Windows taskbar because the calender is drawn below the edit, it should be above it.
TTimeEdit's horizontal position near the right screen edge is correct, but vertically the popup form also overlaps with the Windows taskbar for the same reason.
TTimeEdit has the additional issue that the gray hour bar at the top shows only the second row, probably if the current hour is after noon time. The first row with the hours 0-11 come in by using the mouse wheel (Win 7, standard pixel densitiy, i.e. no high dpi). (see screenshot)
And TTimeEdit raises a SIGSEGV at program exit: Press the time icon to open the time input form, press ALT (I wanted to take a screen shot with ALT-Print). The form closes. But when the program exits, the SIGSEGV is reported.