The Queen does send letters of congratulations - my grandparents got one years ago from her. I think you have to write to Buckingham Palace with all the details.
Hi I have just applied for a diamond wedding telegram for my inlaws. I logged onto the General Registry Office Tel: 0845 603 7788.. You just give them all the details. date of marriage etc. place. full names of both parents and for �3.00 they send the details to Buckingham Palace and hey presto its done.
If you go to the General Registry Office Website you can see the telegrams they can arrange to send via Buck. Palace.