Okay, so we didn't have Mail Order Ponies, but we did have Club Ponies. Yes, there were MLP clubs in lots of countries, but here in the UK we had Club Baby Ponies issued between 1989 and possibly 1992. I'm gonna take this year by year.
The first baby, issued in 1989, was not given a name. Despite club adverts showing her with red hair, she actually was white all over with no symbol, the idea being that the baby could be named and given a symbol by her new owner. This was the first offer of it's kind and the baby was sold to the kids via slogans such as 'I'd love to be yours!' throughout the advertising material.
In 1990, the Baby Pony had a name - Baby Bridesmaid. She came with various activities for the kids to do, and her symbol was metallic silver paint.
1991 introduced Baby Fun & Games, with her bright red hair and red dice symbol.
Well, in recent weeks there has been extensive discussion about these babies on the MLPUK list. Thanks to Tic Tac Toe's information, I can finally positively identify Baby Ribbons and Hearts as the final baby pony's name. Apparently she was offered as a horseshoe points pony only through the My Little Pony club at the end of 1991 - when members of the club received the news that the club was going to end.  This tallies with the fact that surplus of her were sold through Partners stores (at least in the Midlands area) since that is where ours came from. Though she is not a club baby in the same sense as the other club baby ponies, since she was offered through the club as an offer it seems likely that she was intended to be the 1992 club newborn, only the club decided to shut down before this, and so she was offered as a points pony instead. Failing that, the left over stock was sold through Partners (and maybe other stores in other parts of the country) in unmarked bags very cheaply indeed.

To see examples of the application forms and comic advertisments for the pony club, click on the following links...