I'm not sure I understand the question. English bath is quite different from Latin balneum which is pretty straightforwardly borrowed from Greek balaneion. They do both start with b, but Greek balaneion is from the PIE root *guel- which yields German Quelle and English qualm, while Germanic bath, Bad, is from *bhe:-, *bho:- 'warm, roast', which yields Greek phogo 'to roast'.

The city of Bath is a loan translation of Latin Aquae 'waters. baths'.

Ceci n'est pas un seing.