Woefully misled British politicians (3,4)

I believe the answer is:

lib dems

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'politicians' is the definition.
I don't know anything about this answer so I can't tell whether this works.

'woefully misled british' is the wordplay.
I cannot quite see how this works, but
'british' could be 'b' (abbreviation e.g. in 'BBC') and 'b' is found within the answer.
an anagram of 'misled' is 'lidems' which is located in the remaining letters.
This accounts for all the letters.
This explanation may well be incorrect...

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I've seen this clue in The Times.

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