The term dad has origins in children's speech:

recorded from c.1500, but probably much older, from child's speech, nearly universal and probably prehistoric (cf. Welsh tad, Ir. daid, Czech, L., Gk. tata, Lith. tete, Skt. tatah all of the same meaning)

Daddy is the diminutive of this:

The OED quoted here adds more, saying:

Occurs from the 16th c. (or possibly 15th c.), in representations of rustic, humble, or childish speech, in which it may of course have been in use much earlier, though it is not given in the Promptorium or Catholicon, where words of this class occur.

Of the actual origin we have no evidence: but the forms dada, tata, meaning 'father', originating in infantile or childish speech, occur independently in many languages. It has been assumed that our word is taken from Welsh tad, mutated dad, but this is very doubtful; the Welsh is itself merely a word of the same class, which has displaced the original Celtic word for 'father' = Ir. athair.

A childish or familiar word for father: originally ranking with mam for mother, but now less typically childish. Cf. daddy.

?a1500 Chester Pl. (Shaks. Soc.) I. 43 Cayme. I will..Speake with my dadde and mam also..Mamme and dadd, reste you well! [Of uncertain date: the MS. is only of 1592. Harl. MS. of 1607 reads (ii. 678) 'sire and dam', (ii. 681) 'father and mother'.]

1553 Wilson Rhet. 31 Bryngyng forthe a faire child unto you..suche a one as shall call you dad with his swete lispyng wordes.