Hi -- From what I gather you can use a ICP (independent connection provider) of your choice providing they are licenced and accredited with NERS (national elecricity registration scheme) to carry out work on laying new cables and connecting to the national grid.
I enclose a list of qualified contractors with the accreditation showing which areas they cover, it can't harm to contact the nearest companies to you and ask them for quotes, their contact details are to the top right of their individual details box.
A firm called CE electric UK handle new connections for YEDL & NEDL, so EON cannot have the monopoly. I am not surprised they went quiet though when you asked their estimates manager that question. !