If I understand this correctly, Anne's children rank rather highly as grandchildren of the Sovereign. However, once King William V ascends the throne, they will no longer be grandchildren/nephew and niece of the Sovereign and will therefore lose their rank.
Currently, Peter and Zara are ranked
8th and
10th in, respectively, Official Male and Female Orders of Precedence. When Charles becomes King, and assuming we deal with the same people as now (that is to say, excluding new births and deaths), Peter will retain his
8th position (as the Sovereign's nephew), while Zara will actually move up one spot and occupy the
9th position (as the Sovereign's niece). So, no major changes for them. Of course, once William and/or Harry start having kids, they'll get bumped further down the line.
Official Order of Precedence for Men as of now:
1. Prince Philip
2. Prince Charles
3. Prince Andrew
4. Prince Edward
5. Prince William
6. Prince Harry
7. Viscount Severn
8.
Peter Phillips
9. Viscount Linley
10. The Duke of Gloucester
11. The Duke of Kent
12. Prince Michael of Kent
Official Order of Precedence for Women as of now:
1. Elizabeth II
2. The Duchess of Cornwall
3. The Countess of Wessex
4. The Princess Royal
5. The Duchess of Cambridge
6. Autumn Phillips
7. Princess Beatrice
8. Princess Eugenie
9. Lady Louise
10.
Zara Phillips
11. Lady Serena Stanhope
12. Lady Sarah Chatto
13. The Duchess of Gloucester
14. The Duchess of Kent
15. Princess Michael of Kent
16. Dowager Countess of Harewood
17. Princess Alexandra
When Charles becomes King, and again, assuming we deal with exactly the same people as now, the Orders of Precedence will look like this:
Official Order of Precedence for Men during the reign of King Charles:
1. King Charles
2. The Duke of Edinburgh *
3. The Duke of Cornwall (Prince William)
4. Prince Harry
5. The Duke of York **
6. The Earl of Wessex
7. Viscount Severn
8.
Peter Phillips
9. Viscount Linley
* Prince Philip's position would not be automatic but I assume Charles will issue some sort of Proclamation granting Philip pre-eminence over all men but the King.
** Any children William and/or Harry will have (from legitimate marriages) will be ahead of the Duke of York in the Order of Precedence.
Official Order of Precedence for Women during the reign of King Charles:
1. Queen Camilla *
2. The Duchess of Cornwall (Kate)
3. The Countess of Wessex **
4. The Princess Royal
5. Autumn Phillips
6. Princess Beatrice
7. Princess Eugenie
8. Lady Louise
9.
Zara Phillips
10. Lady Serena Stanhope
11. Lady Sarah Chatto
* Whatever her title, as the King's Consort, Camilla will be ranked above all ladies in the Kingdom.
** If Harry is married by the time, his wife will be ahead of Sophie.