Westy
According to the Victoria County History the Walloons were permitted to settle in 1567 after a begrudging nod with strings from the Southampton Gild and Corporation and with consent from Queen Elizabeth through Lord Cecil. This was the time also that they obtained permission to have a place of worship of their own and were granted the use of the God's House chapel, owned, as has already been pointed out, by Queen's College, Oxford.
The chapel had a history that was reasonably settled it seems except at the turn of the eighteenth century when the Church was effectively taken into the Church of England. The VCH does not mention about burials but it does mention that the CofE classed St Julian (or the French Church as it is also known) as a chapel in the parish of Holy Rood. If I were looking for records of the burials I would be searching for the records relating to the host bodies as much as of the church itself. Therefore I would be wanting to see the records of Queen's College in respect of the church; I'd want to see the burial records for Holy Rood, which may be in the Southampton Archive or they may be in the Hampshire record office at Winchester as the diocesan record office.
Southampton's church history is complicated and someone has already mentioned about the 'mother church' being St Mary's, but the walled town included four entire parishes and part of a fifth. The churches for these ancient parishes suffered, as has been explained, in the Blitz with All Saints (the parish that extended well beyond the walls) being destroyed, Holy Rood is now a war memorial, at least the shell of the bombed church is, St John's, one of the original French churches of the Domesday period is gone with the other French Church St Michael's standing proudly still today. I cannot remember the other parish and since it has nothing to do with St Julian I shall not go and look it up. The fact that Holy Rood was so severely damaged, it may be that some registers and records were destroyed.
Good luck in your quest
John