OC 261: CONVERSION AFTER SHABBAT COMMENCES
Conversions may not be done on a Day of Rest (Shabbat or festivals). Nonetheless, if one did convert on Shabbat, the consensus of the authorities in this after-the-fact case is that the convert is fully obligated to observe the remainder of Shabbat in spite of not having been obligated at its commencement.
OC 294: HAVDALA AFTER CONVERTING ON SATURDAY NIGHT
Even if one converted immediately after Shabbat, one is not obligated to bless over ending Shabbat (havdala) since one did not observe a Shabbat so as to need to end it.
OC 370: THE EFFECT ON AN ERUV OF A SHABBAT CONVERSION
The Jerusalem Talmud notes that one who improperly converted on Shabbat morning after dawn “acquires a residency” that then interferes with the shared neighborhood (eruv) of permitted carrying that was set up before Shabbat. Thus, the convert would have to cancel their right to carry in the area in order to allow everyone else to carry in the area in the case of such an eruv.