Most Chinese people I know don't have siblings. They refer to their cousin as "cousin brother / sister".
Is this common?
If so, is this a modern change, perhaps reflecting the one-child policy?
It's not really clear to me what you mean with "cousin brother / sister". Do you mean for example 表姐姐姐 or a 表姐 simply called 姐姐.
To be complete: 表哥 or 表兄 is the older male cousin via female line. 表弟 is the younger male cousin via female line. Similar you have 表姐 and 表妹. Via the male line you similarly have 堂哥 or 堂兄, 堂弟, 堂姐 and 堂妹.
These words are not new.
Nowadays a lot of only children want a 哥哥 of a 姐姐 to "look after them" or to have somebody to whom to can go to when one has some questions or problems and they search for them (for example in the family or a good friend), and afterwards always call them 哥哥/姐姐. Also cousins are often considered as brothers and sisters and are often simply called 哥哥, 弟弟, 姐姐 and 妹妹.
Cousin is 表哥/表姐/堂哥/堂姐, and the 哥s and 姐s do not reflect the one-child policy. "Cousin" just happens to have 哥 and 姐 in them because it makes sense to say that cousins are "siblings" only with different parents.
But a lot of people nowadays refer to their cousins simply as 姐姐 or 哥哥 ("sister" or "brother") probably because they don't have any siblings themselves.
If my cousin's nickname is 豆豆, I would call her 豆豆姐姐.
But you should also be aware you can call those that are slightly older than you 哥哥/姐姐 in informal situations.
If your co-worker's last name is 周, you could call her 周姐. 姐 here has nothing to do with whether you two are related.
The way to call a relative may vary from place to place. The general idea is given by the previous answers. I'll show what we do in my family, which is on the north of Beijing.
Generally three prefixes are used.
"兄弟姐妹" would be added cording to the gender and age of the sibling.
"表哥" and "堂哥" are also valid, but I've never heard "姨哥".
In addition, these prefixes can be dropped when the children are in one "family". The "family" can be rather big. All descendents through the mail line can be considered in one "family". When girls get married, move to another "family" (the way to refer to them in the original "family" will not change). So "堂兄弟姐妹" are in fact not often used, as they are always in one "family", and "哥哥" "姐姐" "弟弟" "妹妹" will used in stead, maybe adding other prefixes to distinguish from each other (e.g. "二哥" "三姐" etc.).