First off, a purely factual - not a moralistic - answer. There is a formal separation between the electoral roll and the council tax register. The first is a legal requirement overseen by the local Electoral Registration Officer. The second determines who is responsible for paying council tax in a particular property. Separate entities.
Not being on the electoral register DOES affect whether you may or may not get a mortgage.
The REAL question you are asking is: can and would any council bother to cross-reference publicly available data (the electoral roll) against its council tax register to see if there are any discrepancies and nail potential offenders?
The answer is that they do. Because the electoral register contains publicly available info, this determines that a council (or any other private or public body) can access it at will.
And they do: Valuebill is a project set up by the Office of the Deputy Prime Minister to match up all the info contained by various local records offices, including council and local business tax records, the electoral register and the local gazetteer (street records). The specific aim of Valuebill is to increase council tax collection rates.
In other words, be afraid, very afraid. Either that or pay your council tax. You have been warned.