What does Agatha Christie’s line, ‘Speak through one’s nose full of b's and d's’ mean?

When one has a cold, with a stopped-up nose, the letters m and n (and v and others) are not nasalized and so come out as b or d or other sounds. Nasalized, non-stopped phonemes can get dropped entirely, like the l in the following sentence.

I hab a co'd so I ca't talk. [I have a cold so I can't talk.]

In the following sentence, the n becomes a d and the m sounds turn into b sounds:

Whed are you beeting be for breakfast? [When are you meeting me for breakfast.]

And so on.