There has never been on-screen complete, intentional nudity in Doctor Who, though there has occasionally been implied nudity off-screen played for laughs. The Third Doctor is seen taking a shower in "Spearhead from Space" and the Eighth Doctor appeared with just a thin sheet covering him and the Tenth Doctor's clone was naked when he generated a body to go with his hand in "Journey's End". The Eleventh Doctor was also nude while changing from his raggedy suit as Amy Pond watched him in 'The Elevanth Hour'. He was also seen in the shower, whilst being a lodger in "The Lodger". The Eleventh Doctor was also painted naked, and shown (implied) naked under a girl's dress in "The Impossible Astronaut". Although in all of these instances no genitalia (penis) was shown.
There have been two known instances of visible breasts in Doctor Who. Leela's nipples were clearly visible through her wet dress at the beginning of "The Talons of Weng-Chiang" Part Four and Amy Pond's nipple was briefly visible (albeit out of focus) down her gown around two minutes into "A Good Man Goes to War".
There have occasionally been instances of full nudity in adult-oriented spin-offs such as Torchwood, examples being "Dead of Night" and "Immortal Sins". Class showed buttocks in "The Coach with the Dragon Tattoo" and also depicted sex in a fairly non-explicit way in "Co-Owner of a Lonely Heart". The spin-off "Zygon: When Being You Just Isn't Enough" (BBV Productions video) also featured penis and breasts.
Don't forget Sergent Benton in "The Time Monster".
Also when jumping through cars in "Gridlock", the Doctor meets a pair who are naked and cover themselves with newspapers.
Also, Larry Nightingale from "Blink" was first introduced being naked.