"1677 James Pawley becomes Jacques Polet after Joining the Threadneedle Street Church with testimony (Could this be similar to an adult baptism ?)"
Hi Akira, if that was him , then the testimony was from Canterburys Walloon & Huguenot Church. So you'd be looking there for his origins rather than in London. Many refugees went there first after arrival from the continent as it was nearer the coast, also between 1580-1675 the Weaving industry was established there, before being largely eclipsed by the London Spitalfields Weaving community from 1675 onwards. Many of those who used the Canterbury Church, were actually Walloons rather than French. The Walloons were Francophone subjects of the Spanish Emperor in the Netherlands, who fled the religious persecution against the Calvinst faith there.
In regards the testimony, when members joined the Threadneedle Street church in London they were required, wherever possible to have a Testimonial signed by the pastor of the church they had come there from, basically confirming they were indeed Protestants, and had knowledge of the basic tenants of their faith (Calvinism). This eased their passage into the new congregation considerably. If they didn't have one they'd either be vouched for by an existing member who knew them, or failing that, they's undergo religious training and testing by a pastor, and when they passed this, the pastor would provide their testimony.
If that was not him joining from Canterbury in 1677, he may have been born in London, but he would not likely have been the James Pawley Born 1650 To Thomas Pawley and Joane at Stepney, London, who a native Londoner. He would have been baptised at the French Church Threadneedle Street.
The only possible candidate I could find was a Jacques Polle son of Jean polle and Jacqueline De la Motte baptised there in 1652. However he was married in 1682 at that church to an Elizabeth Gilson. So is not your man.
There is also a different Jacques Pole, native of Leyden (one of the Dutch territories, now Leiden) marrying an Isabella Smith, of Canterbury, at the French Church Threadneedle Street, London, in 1671.
That couple had three children bapitsed there Sarah 1674, Pierre 1677, Lea 1680.
Since your Jacques was indeed a widow on his remarriage in 1683, I think that is likely him, and he was a refugee born in Leydon. Also given the 1677 testimony from Canterbury, and his wife being born there, I think you need to check the Huguenot records there for more info, possibly on what year he arrived. This does make him a Walloon Calvinist, rather than a French Calvinist (a Huguenot) subtle difference but worth baring in mind.
