I wrote something for an Australian magazine and I was told that the concept of babies being found under the gooseberry bush was not familar in Australia. In fact, the editor told me that she had never seen gooseberries in Australia.
Gooseberries certainly grow in Britain, although you don't see them that often in supermarkets, perhaps because they are fiddly to top and tail so people don't like cooking with them. Gooseberries are not as popular as they used to be, mind you. In the mid-19th century there were 170-odd gooseberry shows a year, now there are eleven, held mostly in Cheshire and Yorkshire in the north of England (there is lots about gooseberries and gooseberry shows here).
I looked up 'gooseberry' in the OED and, although people have in the past suggested that the word comes from a similar Dutch or German word, or is related to the word 'gorse', the Dictionary feels that the word probably comes from the two words goose and berry.
Gooseberry has other meanings. It can mean chaperon, or a third person whose presence is not wanted by a pair of lovers. To play old gooseberry used to mean 'play havoc'. The big gooseberry season is the time of year when newspapers are short of real news so fill their pages with trifles.
No-one seems to know for sure how gooseberry bushes came to be associated with the birth of babies. Gooseberry blossom is symbolic of anticipation, so that might be it. Another possibility is that gooseberry bush was 19th-century slang for 'pubic hair' (see here).
There are regional names for gooseberry eg grozet (Scotland) and feaberry.