Why is the Airbus Beluga XL based upon the A330, and not upon the A350 or A380?
Considering that (one of) the main reason(s) for introduction of larger Beluga XL is to meet the demands of the A350 XWB production, it doesn't make sense to have a modified A350 for the next transporter: Airbus would prefer to sell A350s to customers than to modify them for internal use.
A380 is not supported by airports near most of the Airbus production centers- defeating the very purpose of having the Beluga XL fleet.
In the end, it boils down to which aircraft can satisfy two requirements- payload and runway capability at Broughton. From caan.asia/en:

Airbus wants an aircraft which not only meets high payload capabilities - including capacity to carry two fully-fitted A350 wings - but can operate within airfield landing limitations at its UK wing facility at Broughton, to which it will deliver A350 wing covers. Broughton's declared available landing distance for the shorter runway 04 approach is 1,663m (5,460ft).
Airbus considers this restriction to be too tight for a modified A330-300 or A340-500, while the payload requirement is too high for an A300-600. It believes an A330-200 variant - tentatively designated the A330-200XL - could potentially cope with the landing criteria at projected weights of around 135t, and is the most promising option.

Boeing Dreamlifter, on the other hand, is purpose built for B787 transport- wings and fuselage, while the Beluga transports only the wings. The distances involved in the B787 production (compared to the only European production centers of A350 XWB) means that Airbus can afford to transport only the wings, rather than fuselages too.