'The Quatermass Xperiment', 1955

On August 26th 1955 it was premiered “The Quatermass Xperiment” (aka "The Creeping Unknown") film, in which Jane acts as a little girl but originally was uncredited. 

In 1950s England, professor Quatermass’ manned rocket-ship returns to Earth but two of the astronauts are missing and the only survivor, Victor Caroon (Richard Wordsworth) seems ill and unable to communicate.

Photos 1 to 3) From the net. If they are yours, comment below and I'll give you full credit.
Photo 4) So it Goes... blogspot blog.
Photos 5 to 8) astridllmatumblr (oringally, they are GIFs).
Photos 9 to 11) Just Screenshots Blogspot.
Photo 12) Joseph Black flickr.
Photo 13) Movie dude.
Photo 14) Journeys in Darkness and Light wordpress.
Photo 15) Horror cult films.

The Buccaneers, 1957

Robert Shaw (Captain Dan Tempest), Jane Asher (Jenny Purdy) and Peter Soule (Flip Purdy) in “The Buccaneers” episode “Flip and Jenny” (Season 1, episode 32) aired on April 24th, 1957.
Produced by Sidney Cole, directed by Peter Maxwell and written by Neil R. Collins.

Captain Dan Tempest finds two children, Flip and Jenny Purdy, a young brother and sister, stowed away on his ship. They tell a story of their father, an indentured slave, being thrown in jail for his refusal to work until the local land owner agrees to educate the children. The children convince Captain Tempest and his men to break their father out of jail. 

 
Also starring Robert Hardy (Lord Hinch), Paul Hansard (Taffy), Brian Rawlinson (Gaff Guernsay), Edwin Richfield (crewman Armando), Wilfred Downing (crewman Dickon), Colin Douglas (Papa Purdy), Seymour Green (Constable Herridge), Derek Tansley (Sylvester, gentleman and flute player), Roy Purcell (guard), John Dearth (Bellows), Dawson France (Piggot), Willoughby Gray (crewman Pop), Pearl Prescod (unknown).

Source with video of the episode.

Photos 1 to 4) Free classic TV shows.
Photos 5 to 13) From IMDB.

Mandy, 1952

Five-year-old Jane Asher's first acting role was the part of Nina in the film "Mandy" (also known as "Crash of Silence"), an emotional and well-received story about a deaf-and-dumb girl, Mandy (Mandy Miller), which featured Nancy Price, Phyllis Calvert and Doroty Alison. 

Screeencaps made by me. Please give credit when using them, thanks.

The watermarks are necessary because of screen copying for web use without permission or credits.