In India we consider rivers to be holy and respect them. Sometimes swearing by them to prove a point. But in Namastey London, Bebo Malhotra (an UK based NRI) swears by river Thames to vouch for her daughter's virginity and even compares Thames to Ganga. Is Thames in UK considered to be holy or auspicious? I mean do people swear by Thames in the UK?
I have lived in the UK for over 35 years and I have never heard this expression once. To the best of my knowledge there is no sentiment, emotional or spiritual attached to that (or any other) river.
It is probably a quirk of the speaker. He considers rivers holy but has relocated to the UK so it the famous UK rivers that come to his mind.
I grew up in the London area and never ever heard anyone swearing by the Thames in the manner you describe. It is entirely possible that people of Indian origin have transplanted this behaviour but no local would do so.
Swearing by the Thames, in the sense of cursing while in the vicinity of the river, is rather more common - especially after the pubs close.