Depends on whether you shop at Old Navy solely due to price, or whether that's your particular style/aesthetic preference.
If you're particularly attached to the Old Navy style (by which I guess I mean something pretty sturdily "all-American"), then I guess Gap is the nearest thing, albeit it's obviously pricier (especially if you're buying in the UK).
If you're open to different looks and/or care solely about price, then yeah, everyone else's suggestions are good. Matalan and Primark are dirt-cheap.
If you want stuff a bit trendier but still reasonably priced, Topman and H&M are pretty cheap. I seem to recall Zara is pretty cheap in the UK too (despite being quite pricey in the US). Next and Burton are probably in a similar price range. Depends on your style I suppose.
Also, if you want dirt cheap, big supermarkets like Tesco and Asda also do a clothing range (roughly the equivalent of buying from K-Mart or Walmart).