Hello Elfin
I have a fair bit of experience researching Huguenot ancestors, especially in Londons East End .
First off just to give context, the Huguenots came over to England in several waves, the first after the Massacre of St Bartholomews Eve in the 1500's. This didn't really leave much of a lasting impact, but the real big wave which did came a century or so later in the 1680's, following the Dragonnades and the Revocation of the Edit of Nantes in France, this probably at very least 50,000 strong. Another surge came to England between 1698-1702, maybe 10,000 -15,000, most of these having been refugees already in the Netherlands, which had seen similarly huge numbers flee to them in the 1680's. It then settled down considerably, though steady numbers continued to come at least until Louis XIV death in 1715.
There was then a slightly more favourable condition to Protestants in France, though there faith was still illegal, so not many came in the 1720's/1730's. The Protestants still in France, the 'Desert Church', felt confidant enough to start worshiping openly again around 1745/6 in some areas, in the hope they would be seen to be 'harmless' and the law might then be changed in their favour, but unfortunately this instead precipitated another back lash against them from around 1748, and for next 10 years or so, there was a fresh wave of refugees , nowehere near as big as 1680's, but still maybe between 5,000-10,000, particularly from the Normandy and Poitou provinces where the Roman Catholic Church authorities held considerable sway over the local Intendants.
The Protestants were not finally given civil status in France until the 1780's, and not full equality until under Napoleon at the turn of the 19th C. There were still small amounts of refugees coming to London (perhaps 200 or so) as late as the last decade of the persecutions, the 1770's, almost all these from the Cambresis and Picardy, on the Flemish borders, where again the cause was a particularly zeaolus regional Catholic Clergy, with much influence at local government.
Anyway apologies for the potted history lesson....but I think it is important for you as your family is using the Threadneedle Street Church at an unusual and relatively very late date, the 1790's. This is a very interesting time for the church and the French community in London. Very few of the Refugees from the original waves 1680's-1720's, were still using the church by this late a date, most had become well assimilated by the 1750s'/60's, and if not the son and daughters, then the grandsons and daughters of the original refugees certainly had mostly become Anglicised.
The congregation of Threadneedle Street at the outset of the 1790's largely then consisted of this later wave of refugees from 1748-1780, and their families, and perhaps one or two older members, surviving children of the earlier waves.
The 1790's saw the numbers of baptisms at Threadneedle Street shrink hugely again, and it would never really recover. Partly this was because no new refugees were now coming to replenish the church, but also this was the period when England went to war with Revolutionary France. Londoners have always had a reputation for being somewhat excitable and prone to the mob and rabble (e.g, the attacks on Catholics during the Gordon riots just a decade before this in 1780, and then later the attacks on German Londoners during WW1). To be seen as 'French' during this period was undoubtedly dangerous in London, so a great many Anglicised their names and switched to the English churches. In effect it greatly speeded up a process which had long already been going. By the opening decade of the 1800's only perhaps two dozen or so families still regularly used the French Church at Threadneedle Street.
Your Vasseurs are interesting because they are using the church during the 1790's, but, assuming it is the same couple, they are also using the Anglican churches. (No moving around required for this by the way, they would have most likely been living in Bethnal Green or Spitalfields, and Threadneedle Street, St Leonards, St Matthews and Christ Church were all within easy walking distance, and most Huguenot families will be found using all four of these churches, and possibly a fifth too, St Marys Whitechapel). His wifes surname Baker is English (If it had been for example Boulanger and she had Anglicised it, they would still have used the French form at Threadneedle Street) so he seems to have married a local. I'd think then he is most likely the son of a first generation refugee, who came over 1748-68, and was born in London himself, or he was bought here as a very young child at the tail end of this spectrum.
You would expect to find their names recorded with the French forms in the French Churches, and English in the English Churches, even well before this later date, this was just the protocol for recording their names. For example a Jacques will be a James in English churches, Francois a Francis, Andre an Andrew, and so on. This is not at all unusual and doesn't really reflect what they themselves prefered to be known as day to day, so I wouldn't draw any firm conclusions from that.
I have a few of the Huguenot Societies publiciations and will look through them and see if the Vasseurs are mentioned and get back to you if I find anything of interest, but hope that helps a bit with your searches for now.
Richard