Hi Mgr, have just added some details re the marriages baptisms etc to my previous post.
Personally I would think this is likely a Walloon family. You will hear some people talk about all French speaking refugees before a certain date being Walloons, and all those after a certain date being Huguenot, but this is nonesense.
There were Huguenots coming to London almost from the start of the reformation in France in the 1530's, before Walloons, and they continued to come, especially at times of distress in France the religious wars 1560-1600, the seige of La Rochelle in 1620's, then a gentle trickle started in the early years of Louis XIV's reign eventually becoming a flood in his later years.
This of course overlaps with the time the Walloons were coming roughly 1550's-1680. So without having an absolute place of origin, it is impossible to state for certain, whether they were Walloon or French. But it is true that before around 1650, Walloons were more numerous than Huguenots here, because the persecution they faced under the Spanish was more intense, I think the Inquisition was burning thousands at the stake there at one point, worst persecution anywhere in Europe. This alone would incline one to think your Pollets are Walloon.
Then there is also the evidence of Akiras family. They were certainly Walloons, coming from Tourcoing, which showed the surname had a presence in the Walloon lands.
It may even be possible Akira relatives are linked to yours, with your branch leaving their hometown in the 1580's/90's, and Akira's either remaining, or fleeing at same time, to more northern most towns in the Spanish Netherlands, such as Brussels, where the persecutions were less intense, before fleeing again to Leiden in 1633, when the persecutions began in these towns too, during the thirty years war (1618-1648).
So to cut a long story short...the evidence points towards Walloons, but you'd need confirmations of their place of origin (may be found in the Canterbury Church admittance records?). Of course some would argue the difference is fairly academic, as they were both Francophone in language and both Calvinist in faith, which is why they so readily shared churches in England, and the divisions soon melted away once here in any case. Many Walloons today want to be part of France not Belgium, and since Louis XIV annexed some Walloon land, some indeed already are, including the Wallons of Tourcoing today. But historically the Walloon people and lands were never, up until then, subjects of the French Crown, but rather the German/Spanish Holy Roman Emperors.