What are the connotations of "clueless"?

I agree that the term is not nearly as grievous an insult in the US as it apparently is in the UK, and can be used in situations where oblivious might also be used.

As the question is about some fairly subtle particulars of usage, and existing dictionary definitions aren't helpful in distinguishing between these usages, I will take the liberty of constructing a working definition to help structure the rest of my answer:

Clueless, adj.

Lacking knowledge or comprehension

I. In a general domain

A. Due to incapacity
B. Due to lack of experience with or exposure to existing evidence

II. Of a specific fact, usually due to lack of evidence

In Sense I, cluelessness about a subject may result from either a (reparable) lack of experience or from an inability to comprehend. However, even in the more severe, lack-of-capacity case the person does not necessarily lack capacity in other areas of life and learning. Examples of this kind of usage:

It's also very hilly, which should have been a clue for me about how difficult this race was going to be, but as I said, I was clueless on all aspects. The race only draws about 250 participants, and again, I was clueless as to why, until it was too late to turn back.
—Ileta Coley, "Blood is Thicker than Water", First Marathons: Personal Encounters with the 26.2-mile Monster, ed. Gail W. Kislevitz, 1999

Despite himself, Dan laughed. He turned around and looked at me affectionately. “You're the smartest woman I know, but you're completely clueless about math, aren't you.” “Guilty as charged.” I grinned.
—Janice Kaplan, A Job to Kill For: A Lacy Fields Mystery, 2008

Although this usage is not necessarily horribly insulting, neither is it quite the same as oblivious. However, the second sense comes much closer to being synonymous with oblivious.

In sense II, clueless can mean that the person lacks knowledge or awareness of a specific circumstance, which can be due to factors entirely beyond the individual's control. I would say that this at least comes very close to synonymy with oblivious. For example:

“I'm under the impression you were clueless about the affair?” Trevor once again offers a nod and states, “No sir. I had no idea about the affair or the fact she was pregnant.”
—Stephen Mitchell, The Forgotten House, 2013 (Trevor was oblivious to his wife's affair)

And make no mistake...you will be judged and judged heartlessly by those who are clueless as to what your life has been like as a caregiver.
—Sandra Savell, Dear Clueless, 2015 (Non-caregivers are likely to be oblivious to the realities of life as a caregiver)

Suzie could see the steady crisscross traffic up ahead—late-night motorists, coming and going, clueless to the danger speeding their way.
—David DeLee, With Intent to Deceive, 2014 (The motorists were oblivious to the danger speeding their way)

The following quote illustrates rather well some of the subtleties of the term's usage:

Dr. Falmer was just as clueless as he'd hoped she would be, but it wasn't the right kind of clueless. She wasn't a snob, intellectual or social. She wasn't a fool, or an airhead like Marcey and Arrow. She was just a pleasant, well-meaning, quietly dressed woman who probably didn't have the faintest idea what she'd gotten herself into. Carl Frank hadn't wanted to be the one to let her in on the secret.
—Jane Haddam, Cheating at Solitaire: A Gregor Demarkian Novel, 2008

The relative "insulting-ness" of the term might be judged to some extent by how often it is applied in the first person. Looking only at the frequency of use within the American English corpus (here) and British English corpus (here) on Google Ngrams, we see that "I was clueless" is five times as common for US writers as for British writers. Of course standard disclaimers about Ngrams apply, but the numbers are at least suggestive.

I don't think that this belongs in the body of the answer, but in looking for clues to differences in usage I found that the most common word to precede clueless in the American corpus is was, and following the path of most common following words it appears that the most common phrase containing the word is likely was clueless about how to. To me, this phrase is suggestive of a lack of knowledge that can be rectified. In the British corpus, on the other hand, the most common word to precede clueless is a; a clueless___ looks very much as if it is going to be an insult.

(To find this, I used a wildcard search to find the most common word preceding clueless in each corpus—"* clueless"—and then added wildcards to the end of the most common phrase until Ngram returned no answers. That is, "was clueless *" in American English gives "was clueless about" as most common, etc. until "was clueless about how to *" gives no results. In the British English corpus, "a clueless *" is the dead end.)

I don't believe that they are synonymous. Someone with no knowledge of domestic plumbing might be clueless when it came to stopping water running through his house from a failed pipe, but he certainly would not be oblivious to it. A fully qualified and experienced plumber, however, might be oblivious to a leak under his floorboards even though he was quite capable of fixing it if he knew about it.