No.It said on this evenings BBC news the new rules are not retrospective. So people like princess Anne will stay the same. I would think we will get some bad language from her!
It depends whether the changes are retroactive, or not.
1. If the changes are retroactive (starting from Queen Elizabeth's descendants), then Princess Anne will indeed move higher in the Line of Succession; she will be immediately after Prince Charles and his descendants, and above Prince Andrew and Edward, and their descendants.
The retroactive change would also mean Lady Louise would be above her younger brother, Viscount Severn, in the Line of Succession.
The Line of Succession would look like this:
Prince Charles ---> Prince William ---> Prince William's children (in order of birth) ---> Prince Harry ---> Prince Harry's children (in order of birth) ---> Princess Anne ---> Peter Phillips ---> Peter's' children (in order of birth) ---> Zara Phillips ---> Zara's children (in order of birth) ---> Prince Andrew ---> Princess Beatrice ---> Princess Eugenie ---> Prince Edward ---> Lady Louise ---> Viscount Severn
2. If the changes are not retroactive (they come into effect starting with the future generations), then Princess Anne would remain in her place.
Prince William and Prince Harry's children, however, will be ranked in the succession line according to their birth order, irregardless of gender.
The Line of Succession would look like this:
Prince Charles ---> Prince William ---> Prince William's children (in order of birth) ---> Prince Harry ---> Prince Harry's children (in order of birth) ---> Prince Andrew ---> Princess Beatrice ---> Princess Eugenie ---> Prince Edward ---> Viscount Severn ---> Lady Louise ---> Princess Anne ---> Peter Phillips ---> Peter's' children (in order of birth) ---> Zara Phillips ---> Zara's children (in order of birth)
3. Even assuming the changes are retroactive and Princess Anne does move up the Line of Succession, her children would still have no titles as titles are passed down the male, not female line.