That doesn't make any sense to me. Can you post exactly the sequence of statements if Bernard speaks first?

Simply reordering the questions does not make sense, so I can't see it happening any way other than what I posted, which shows that it wouldn't change anything.

It really depends on who says what, but here a version of the problem where Bernard speaks first:

Bernard: I don't know when Cheryl's birthday is.

Albert: I don't know when Cheryl's birthday is.

Bernard: I know when Cheryl's birthday is now.

Albert: Then I also know when Cheryl's birthday is.

That breaks the whole problem because you've removed the part where Albert says something that gives Bernard the information he needs to work out the date.

I don't think it's really a good critique of a logic problem to say, "but if you remove this extremely important part, it doesn't work anymore"!

The conversation would need to be:

Bernard: I don't know when Cheryl's birthday is

Albert: I don't know when Cheryl's birthday is, and I knew that Bernard didn't know.

Bernard: Then I know when Cheryl's birthday is now.

Albert: Then I also know when Cheryl's birthday is.

Then it works. You only have to assume that the parties are logical and tell the truth.

You have to make quite a few more assumptions as well. If they were logical and told the truth, the conversation would be more like:

Bernard: Cheryl told me it's the 17th.

Albert: Cheryl told me it's in August.