Phydeaux3 found that his Backlinks stopped working after one such update because he had hacked the Blog widget. I checked my blog and it was the same case too. It's a good idea to check your blog also (specifically if you applied peekaboo posts hack). If you have the problem, here's the fix that worked for me.
Search for 'backlinks-container' in your template to find this portion of code and add the lines in red. This is the update from Blogger that didn't make it to my template.
<div id='backlinks-container'>In future, I will keep an eye on such updates and inform you if you need to make any change.
<div expr:id='data:widget.instanceId + "_backlinks-container"'>
<b:include data='post' name='backlinks'/>
UPDATE: If you downloaded one of my hacked templates, this change is already incorporated in it.