If one is in a rush when is the earliest one can take off his Rashi Tefillin and change to Rabeinu Tam’s?
It has become common practice not to remove one’s Tefillin until after the recitation of Kedushah in the passage beginning U’Va LeTziyon.
Some authorities maintain that according to the Kabbalah one must [respond to] three Kaddishim while wearing Tefillin (the third being Kaddish Tiskabel). Therefore one should not remove his Tefillin until after the Kaddish which follows the passage beginning U’Va LeTziyon.
The Arizal would not remove his Tefillin until after [Ve]Al Kein Nekaveh Lecha. Some people do not remove them until after the Mourners’ Kaddish.
The Chabad Minhag is not to remove the Tefilin until after the daily Tehillim.
שוע”ר סי’ כה סעי’ לז וספר המנהגים ע’ 5.
Do these spots still apply even if you switch to Rabbeinu Tam? Or for example can Kaddeishim be in Rabbeinu Tam Tefillin?
Yes, the Minhag is to put on the Tefilin of Rabeinu Tam after davening while the Chiyuv is to daven in Rashi’s Tefilin.
ראה שוע”ר סי’ כז סעי’ ה ובסדור ד”ה כל אשר.