UK Constitution is not a 'Book' but rather a collection of traditions and legal rules and the rules are quite confusing. It is an evolvement over hundreds of years. With its 'flexibility', one will argue that it will cause uncertainty and too much power to the legal system which is not democratic. The Hansard poll which found that 54% wanted a strong leader to 'break the rules' was a concern and that potentially has brought out an idea that the whole system needs to be rethought could be a nightmare. Also, does Referandum really suggest a sane and healthy system or will it further divide society?
It would be a gamble to find consensus and hand over to public to pen the constitution.