From the information in the London Huguenot Library the indications are that the name Presant is not Huguenot, but Walloon, that is French speaking rather than Flemish speaking  from Flanders in what is now Belgium.  The distinction is that the Walloons began to escape to England early in the 16th cent to get away from religious persecution by Catholic Spain, which then controlled  what is now Holland and Belgium.   The Huguenots were specifically from France and fled after religious toleration of Protestants was withdrawn by Louis XIV's Revocation of the Edict of Nantes in 1685.  There is no mention of the name in the standard work on the Huguenots called "La France Protestante" by E Haag, which indicates that the name was not Huguenot.

They were granted a disused church as a Cloth Hall, but by 1631 the textile trade had flourished sufficiently for them to need a larger cloth hall and they were moved to a new hall and the church was granted to the Walloons for divine worship in French and all the church records of baptisms and marriages as well as some wills were also in French.  Walloons from other settlements seem to have gravitated there later.   Strangely, very few of the French refugees at the time of the Revocation of the Edict on Nantes settled in Norwich, so the church was always Walloon rather than French Huguenot in its congregation.    Gradually as the people assimilated support for their own church faded and it was in trouble by 1669, but managed to survive until the church was finally closed in 1832.   The church was St. Mary the Lesser at Tombeland, Norwich. There was large settlement in Yarmouth of Protestant refugees from Flanders in the early 16 century and settlements of Walloons in various towns in England from 1540 onwards, many of whom settled in London, Canterbury in county of Kent and Colchester in the county of Essex.   Some later moved from London and Canterbury to Norwich.   The settlements at Canterbury and Norwich were known for their weaving and at Yarmouth for fishing and cooperage.

In looking at the references to the family name, one has to remember that: spelling was pretty optional, for example, I think that Shakespeare varied the spelling of his name himself at different times in his life. And the records were written by a clerk trying to put down phonetically what he heard so that the French pronunciation of Presant would be Pres + a nasal 'n' sound. 

From the Register at the French church at Canterbury:
1694 baptism of Jean, fils d'Abraham PRESAIN et Susanne Lecluse
1718 baptism of Anne, father Pierre Graard and mother Judith PRESEN
1602 marriage contract between Simon Ferret and Isabeau Six, widow of the late Antoine PRESIN
Sponsors mentioned in baptism records 1627 Marie PRESIN   1632 Jean PRESIN

From the Register of the French church in Threadneedles St. London
Sponsors mentioned in baptism records.  1637 Jenne Hainqueau, widow of Nicolas PRESEAU, native of Valenciennes (Flanders). 1682 Jeanne PRESEAU  1684 Jean PRESSON  1688 Jean PRESEN  1697 Jeanne PRESSON

From the Register of the Walloon church in Norwich 1803 mention of the will of Tester PROSANT of Kings Lynn

Before the times of immigration controls and green cards the English Government from time to time would require anyone who knew of a foreigner living in the parish to report it and a List of Aliens in London was taken in 1582 in which the London Company of Weavers reported that they had admitted as a member "a foreign journeyman" (fully trained) weaver named John PRESENT.  The fact that he is reported as an alien suggests that he was still unassimilated, though clearly pretty well established in the textile trade in London.   The Company controlled entry into the trade and governed all aspects by setting standards and sometimes fixing prices.  There was a period when the Norwich weavers were doing so well that the London company could not stand the competition and for a time banned the sale of Norwich textiles in London. "

From the book "A History of Norwich " by Frank Meeres;

There has always been movement of people between Norwich and Europe.  In the 1560's the Netherlands was a colony of Catholic Spain and the governor, the Duke of Alba, was persecuting Protestants in the colony.  In 1565 the city authorities arranged for 30 households of religious refugees from the Netherlands to settle in the city.  Six of the households were  French speaking Walloons and the other 24 spoke Dutch, or Flemish.  More rrefugees followed:  a return of aliens made in 1568 and preserved in the Norwich city archives lists 1,132 Flemings and 339 Walloons in the city.  By 1579 there were 6,000 'strangers' in a population of about 16,000--so that they made up over a third of the population, the largest percentage in any town in England, although the actual number of immigrants settling in London was greater.  Most of all the immigrants were weavers.  They were organized by a committee known as the Governors of the Drapery and by their church leaders.  They appointed 'politic men' (eight Dutchmen and four Walloons) to be their representatives in dealing with the English.  Some followed other trades and professions.  The Dutch community worshipped in the Blackfirars church in St.Andrew's Hall, the French in the Bishop's Palace until 1637, then in the unused church of St.Mary the Less in Queen Street until 1820.