what links 'Blue Peter' with sci-fi classic 'Doctor Who'?

The answers are...

(1) Peter Purves (Presenter 1967-78) played a tourist in black-and-white story "The Chase" (1965)
and went on to play companion Steven Taylor from 1965-66;

(2) Janet Ellis (Presenter 1983-87) played Teka in "The Horns Of Nimon" (1979)

(3) Christopher Wenner (Presenter 1978-80) had a walk-on part in "The Awakening" (1984);

(4) Sarah Greene (Presenter 1980-83) played Varne in "Attack Of The Cybermen" (1985);

(5) Matt Baker (Presenter 1999-2006) played himself in "Aliens Of London" (2005);

(6) Gethin Jones (Presenter 2005-08) appeared as a Cyberman in "Rise Of The Cybermen" (2006, uncredited)
and was also a Dalek Operator in "Journey's End" (2008, again uncredited).

(7) the celebratory BBC2 special "Blue Peter At 50" (2008) was narrated by the 4th Doctor, Tom Baker.

Plus, before Ellis joined "Blue Peter" and Sylvester McCoy became the 7th Doctor,
both worked together on the Children's BBC programme "Jigsaw" (1979-84).

Interestingly, after her tenure as DW companion Ace (1987-89) came to an end,
Sophie Aldred auditioned to be a B.P. presenter - but was unsuccessful.
Then again, Aldred has later suggested this was just as well,
as her career might have ground to a halt if she went from DW companion to BP host!

A notable difference (of many) between "Blue Peter" and "Doctor Who"
is that a few presenters were let go when their contracts came up for renewal
(Leila Williams in 1962, Wenner in 1980, and the late Michael Sundin in 1985)
and one host had his contract terminated in mid-run (Richard Bacon, in 1998) -
whilst only Time Lord was fired, and that was the 6th Doctor, Colin Baker.