I can give you some information about the Lechmere-Tompkins connection.Luct Lechmere was the second wife of Henry Tompkins of Weston Turville, in Buckinghamshire.He was a London barrister (admitted to the Middle Temple in 1755, called to the bar in 1771, a bencher of the Middle Temple in 1792) who became a figure of some local importance in Buckinghamshire in the second half of the 18C - he was its Lord-Lieutenant in 1782, its Receiver-General of the Land Tax in 1784, the Lieutenant-Colonel of its militia regiment.
He married Lucy Lechmere in 1794 (having obtained a Vicar-General's marriage licence on 12 July 1794) and died soon afterwards, in the same year (one can't help speculating whether Lucy was an 18C Anna Nicole Smith!).Administration of his estate was granted to his widow Lucy (on 14 MNar 1795, in the PCC), but the heir to the Weston Turville estate seems to have been his son John William Tompkins (his son by his first wife - unsurprisingly he doesn't seem to have had any children by Lucy).John William died in 1800 (while on service in Ireland as Lt Col of the 6th Dragoon Guards) and left the Weston estate to his step-mother Lucy, so presumably any speculation about gold-digging would be unjustified. (His will was proved in the PCC on 20 Sept 1800)
Lucy seems to have been living at Weston when her step-son died, and stayed there till her death in 1832 or 1833.Her will was proved in the PCC on 29 Jan 1833 (ref PROB 11/1811, folio 22).In it she left her plate and books to her nephew John Lechmere (son of her deceased brother William) and the Weston Turville estate to a Nicholas Lechmere, then serving in the East Indies (ie India) in the East India Company's Artillery, whom she described as the "reputed son" (an illegitimate son, perhaps?) of Nicholas Lechmere, late of Jamaica, esq., then deceased.
At this point I think you finally get the information you have waded through this long posting to find.It seems your Nicholas born 1755 went to Jamaica (and probably died there - you might find his will in the PCC) and his son, your Nicholas , was a gunner (presumably an officer) in India in the HEIC artillery (you might find more about him in the records of the Honourable East India Company).
I don't think Nicholas  kept any link to Weston - certainly the manor house and estate were sold to the Duke of Buckingham in 1835.
Lucy's will also mentioned her sister Elizabeth Tudor.
I hope that has all been helpful.If it has, there is something which you could do for me in return.I think the connection between the Lechmeres and the Weston Tompkinses goes back a generation or two before Henry and Lucy and, if I'm right, that connection will provide the clue to the origins of the Weston Tompkins family, which have hitherto been unknown.In a moment I'll post a new message to the Lechmere Genforum on this question - if you can help with it I'll be very grateful.