In which tradition Om Ma Ni Pad Me Hum mantra is used Mahayana or Thervada? It is Tibetan mantra.
In which traditon Om Ma Ni Pad Me Hum mantra is used Mahayana or Thervada?
Or it is used in all three traditions, since this is Buddhism teaching?
The back cover of the book "The Origins of Om Manipadme Hum: A Study of the Karandavyuha Sutra" by Alexander Studholme states:


  Om Manipadme Hum, perhaps the most well-known and most widely used of
  all Buddhist mantras, lies at the heart of the Tibetan system and is
  cherished by both laymen and lama alike. This book presents a new
  interpretation of the meaning of Om Manipadme Hum, and includes a
  detailed, annotated precis of Karandavyuha Sutra, opening up this
  important work to a wider audience.
  
  The earliest textual source is the
  Karandavyuha Sutra, which describes both the compassion of
  Avalokitesvara, the bodhisattva whole power the mantra invokes, and
  the mythical tale of the search and discovery of the mantra. Through a
  detailed analysis of this sutra, Studholme explores the historical and
  doctrinal forces behind the appearance of Om Manipadme Hum in India at
  around the middle of the first millennium c.e.
  
  He argues that the
  Karandavyuha Sutra has close affinities to non-Buddhist puranic
  literature, and that the conception of Avalokitesvara and his
  six-syllable mantra is influenced by the conception of the Hindu deity
  Siva and his five-syllable mantra Namah Sivaya. The Karandavyuha Sutra
  reflects historical situation in which the Buddhist monastic
  establishment was coming into contact with Buddhist tantric
  practitioners, themselves influenced by Saivite practitioners.


The mantra "OM mani padme hum" seems to be found only in Tibetan Buddhism.

However, Avalokiteshvara can be found in other non-Tibetan Mahayana traditions like Pure Land Buddhism. Avalokiteshvara is officially not found in the Pali Canon of Theravada Buddhism, even if you can find images of him in some Theravada temples.

Mantras or some kind of phrase for chanting can be found in almost all Buddhist traditions like "namo amitabha buddha" in Pure Land Buddhism, "nam myoho renge kyo" in Nichiren Buddhism and "namo tassa bhagavato arahato samma sambuddhassa" in Theravada Buddhism.