James,
If Diana, Princess of Wales was alive today, I firmly believe that she would have married Dodi Al Fayed. Consequently (and despite common misconception), Diana would be Mrs Dodi Al Fayed and loose the Princess of Wales STYLE (as happened to her mother Frances, Countess Spencer who became The Rt Hon Mrs Peter Shand Kydd). The likelyhood is that she would have had a child and would have converted to Islam (as per the usual situation with the wives of Islamic men) and would have had an Islamic child, (Christian name) Al Fayed. I'm not suggesting that she would have started wearing burkhas or anything like that but she would certainly have converted if nothing else if she had married Dodi.
I also think there would be a major political upheaval as a result of any such marriage. The idea that the mother of the future King and Head of the Church of England being a Muslim would have caused problems, especially when William becomes King and once again she would have found herself in every international newspaper but this time for different reasons. I don't think Diana would have been allowed to remain at Kensington Palace following any re-marriage not just to a muslim but particularly to a muslim and having converted herself.
Then there's Charles and Camilla. There would have been two conflicting possibilities depending on how delicate Diana's re-marriage was handled. If it was handled delicately and with little fuss (as would have been likely), Charles and Camilla would have married anyway but much later (possibly this year) and with Diana holding a new marital name, it would almost certainly have made it easier for Camilla to use the title Princess of Wales. Consequently she would have adopted the title HRH The Princess of Wales and held second position in all situations after the Queen in the order of precedence.
If the situation had been handled indelicately, Charles and Camilla would probably not have married and Camilla would still be Mrs Parker-Bowles. The mood of the people and particularly, the mood of the nation towards Camilla while Diana was alive would have remained the same, if not worse. If Diana had not died, she would probably still be doing humanitarian work, the land mine situation would have improved with her patronage and Diana would probably have moved with Dodi to Paris as she herself implied that she would do. The reason for this would be to minimise paparazzi intrusion into her private life.