Not really (at least for the US -- I don't know about the UK). The unemployment rate is only low because so many people have given up on looking for work. The labor participation rate is at its lowest level since 1977. Here is a nice chart:
Also, full employment usually implies rising wages as employers chase scarce talent. This has not been very visible in the US. Perhaps the ability for employers to invest in the global talent pool could be tempering this affect.
If the market were really tight as trumpeted by the media, there would have been wage pressures already. Perhaps the tight market is limited to SV - I certainly see no signs of hot labor market in Boston. Smaller software companies do have quite a few openings, but are very price-sensitive (and picky, too). Bigger companies are mostly shedding people (either through attrition or selective layoffs).