Following answer may help you
perform your date ordering by your date identifier but use to_char() function in select clause and use some other identifier in select clause for date
e.g.
SELECT TO_CHAR(DISPDATE1,'DD/MM/YYYY') AS DISPDATE,
SUM(APPLCOUNT) AS APPLIED,
SUM(CONFCOUNT) AS CONFIRMED
FROM
(
SELECT COUNT(ID) AS APPLCOUNT,
0 AS CONFCOUNT,
STUDENT.APPLIED_ON AS DISPDATE1
FROM STUDENT
WHERE STUDENT.ID = P_ID
GROUP BY STUDENT.APPLIED_ON
UNION
SELECT 0 AS APPLCOUNT,
COUNT(ID) AS CONFCOUNT,
STUDENT.CONFIRMED_ON AS DISPDATE1
FROM STUDENT
WHERE STUDENT.ID = P_ID
GROUP BY STUDENT.CONFIRMED_ON
)
GROUP BY DISPDATE1
ORDER BY DISPDATE1;