What happens to tube nucleus of the pollen tube after fertilisation?
Short Answer: It disintegrates.

Background: Although I was unable to find any primary reference, there were many secondary references stating that the pollen tube does disintegrate. In very short, after the pollen tube reaches the female gametophyte, the pollen tube nucleus disintegrates. For the 'why', it disintegrates to let the two sperm cells pass through, and also since its function of controlling the growth of pollen tube has finished.



For citation, you can have a look at:


Fertilization - Wikipedia
Advanced Biology -
A new mainstream text for the new specifications
Advanced Science; Michael Kent, 2000
Sexual Reproduction in Angiosperms; Kimball's Biology Pages
BIOPL 2410 Introductory Botany; KNiklas, TSilva; Cornell University
The Structure and Functions of Flowers - LeavingBio.net (image source)