What connotation exactly does the word "noddy" have in British English?

Solution 1:

In today's British English you can hear noddy applied to a question or a problem.

Q: I've got a question: when my phone line is down, is my Internet connection down as well? I know it's a noddy question!

You can even hear, speaking of someone:

He's quite noddy. Don't ask him too much!

The meaning is "simpleton". It's not that of fool as in "insane".
This is confirmed by the main entry (there are no less than 5 for the common noun) in the OED:

noddy, n. A fool, simpleton, noodle.

Here are a few quotes (including your own quote).

1530 J. Heywood Love (Brandl) 798 Why, where the deuyll is this horeson nody?    
1550 Bale Apol. 30 b, O beastly nody wythoute brayne.    
1580 Lupton Sivquila 14 Mighte not he bee counted a verye noddy, that woulde pay suche a fine for a Farme?    
1621 Burton Anat. Mel. i. ii. iv. iv, Soft fellows, stark noddies, and such as were foolish.    
1648 Gage West Ind. 101 In his carriage and experience in the World a simple noddy.    
1682 N. O. tr. Boileau's Lutrin iii. 94 And there they sneaking stand like baffled Noddies.    
1705 Hickeringill Priest-cr. ii. iii. 36 The cringing old Noddies and Cathedral-Men, that adore unlighted Candles at the Altar.    
1794 Wolcot (P. Pindar) Sun & Peacock Wks. 1812 III. 265 To credit such a tale I'm not the noddy.    
1838 Dickens Nich. Nick. vii, To think that I should be such a noddy!
1871 B. Taylor Faust (1875) I. xxii. 197 A gray and wrinkled noddy.

Solution 2:

Merriam-Webster defines it as “a stupid person”, while the New Oxford American Dictionary gas “(dated) a silly or foolish person (esp. as a general term of abuse)”. The connotation is really that of foolish, but I wouldn't use it nowadays.

The two sources differ on the origin, though neither is categorical: for Merriam-Webster, it is “probably short for obsolete noddypoll, alteration of hoddypoll (fumbling inept person)”, while NOAD says “perhaps from the verb nod + -y”.

Solution 3:

The use of Noddy as a fool is probably quite rare, at least in any bits of England I've lived in. More common is Noddy as simple or easy - eg. "a Noddy question on an exam"

Solution 4:

The etymology is well described above, but noddy in modern parlance is probably more likely to evoke "Make Way for Noddy," a children's show about a creepy looking puppet that walks.