Anorak on..
Air Arabia is getting some good press here. And Sharjah to Dubai is only a US$15 taxi ride. Can take a while though - the traffic is awful in the morning. Need to allow at least a couple of hours..
A taxi from Sharjah to Abu Dhabi would be around US$50 but the Etihad flights from there to London are often substantially cheaper than carriers from Dubai. Did that a couple of months ago and they were pretty good actually.
Etihad run a coach (nice bus) from Dubai down to Abu Dhabi (it can be cheaper to book Etihad from DXB - London than from Abu Dhabi to London). They also fly to both LGW and LHR which may be useful?
Hope that helps
CS