As you don't need any particular degree for consulting, why not try applying for the graduate schemes like anyone else? All other things being equal you won't be any worse placed than someone with a degree in e.g. history.
You may however have missed the boat for this year.
I would also suggest that you should not be put off by one poor experience - I assume you worked in one organisation for your sandwich year? The construction industry is large (~10% GDP) and extremely varied and it is quite likely that there are some areas and firms that would suit you very well and which you would enjoy very much.
If you are set on consulting then there are of course firms which specialise in project management, which might be a good starting point for your investigations.