I believe the only universities in the UK which do specifically general engineering are Cambridge (but Oxford may also do it), Durham, Aberdeen, Imperial College London, and maybe a few others that I don't remember (I applied for general engineering).
Scottish universities however have a system where you take several subjects in year 1 and then narrow them down so although its not a general engineering course, you could apply for civil and then take electrical and mechanical as additional subjects then continue your degree in whichever discipline you prefer.
I personally applied to Cambridge, Durham, Aberdeen, Strathclyde (for civil), and Swansea (for civil). Got rejected from Cambridge; firmed Durham and insuranced Aberdeen.