There is a reason why I believe that Andrew would have to be a widower to marry again with his mother's permission.
Anne remarried in the church of Scotland as a divorcee is common knowledge. Its also common knowledge that if she had a child from the second marriage,any issue of the female line, the succession to the throne is not an issue.
Its true Charles and Camille married civilly and the marriage was blessed by the CoE with the Queen in attendance. But there again you have to look at the ages of the two that married. I believe it was Charles at 57 and Camilla at 58? Once again, although remotely possible, the chances of a male child being born to this couple and in line for the throne would be a bookie's dream. If I do recall correctly, Camilla did have a hysterectomy around that time period also.. so scratch all bets.
With Andrew its totally different. Although he is divorced and his first wife is very much alive and as we all know... still in his life, the chances for the Queen to give permission for him to marry again I believe would be very slim. Why? What's the difference between Anne, Charles and Andrew? A big one. He could marry lets say for a lack of an example.. Paris Hilton. And they could and might produce a son. That son would be in line for the throne. <My grandfather was 57 when my father was born and near 60 when my uncle was born>... talk about randy old goats!
Andrew's son then by his second marriage would go ahead in the line for the throne over the girls he had by his first wife. As I don't believe the CoE will marry a royal divorced man to anyone (jump in here and correct me.. I don't know the CoE at all) , wouldn't that put the son by the second wife in a position where it would be questioned?
I know I ramble but this is why I suggested that Andrew would never get his mother's permission to marry again unless he was a widower.