Nottingham? If you take that, you'll take anything.

The dating of Friar Tuck is important- the friars were originally mendicant preaching orders- they lived by begging as they travelled round, expounding religion in places where the official religious provision was often very scanty. Originally somewhat informal, they later became regularised as the Franciscans, Dominicans, Auguistinians, carmelites etc. But if the traditional cast includes King Richard I and King John, that's a little early to have generated an outlaw from these formalised orders.

Over on the Continent, there were lay groups who dedicated themselves to semi- monastic lives, often living together in town houses that operated as religious commmunes, known as the Beguines or Beghards. Being out of the control of the abbots and bishops, they were often under the suspicion of heresy, indeed they were later suppressed by the Inquisitions (usually operated by Dominicans and Franciscans).

Also Tuck sounds more like a surname (they were just beginning to gel in England then) than a personal name, whereas a vocational cleric would be Brother Something-Like-Joseph, a religious name.

So my bet would be for Friar Tuck to be an educated and religious layman, perhaps fired with a sense of religious fervour and hatred of corruption and injustice, and in danger of arrest for heresy.