This is a difficult question to answer for all countries except Saudi Arabia, due to the paucity of public opinion polling on said countries and (in the case of the United States) an often low level of geographic knowledge among the US population, as demonstrated in a number of polls. Still, here is what polling suggests.
To start with, we can look at Gallup's list of country-level polls administered in the United States. Oman, the UAE, Qatar, and Bahrain appear to be missing altogether, suggesting that Gallup never polled for them. We see that the majority of USA residents polled had either somewhat positive (41%) or somewhat negative (30%) views of Kuwait. However, this was a single survey from nearly two decades ago (2003)!
Polling for Saudi Arabia is more regular and thus more reliable. Saudi Arabia has had a generally negative perception in the United States since 2002. As of February 2022, 64% of the United States of America had a negative view of Saudi Arabia, whereas 33% had a positive view. Of note, these percentages have oscillated very little since 2002, perhaps in line with the relative lack of change in the Saudi system of government since that time period. Of course, another possible influence is Islamaphobia: several Muslim-majority countries saw an increase in their disapproval ratings in the United States in the wake of the 9/11 attacks. The increase for Saudi Arabia was the highest, however: this may be attributable to the fact that the majority of the participants in the attacks were Saudi nationals, and this would have been the context in which most Americans would have been hearing about Saudi Arabia in the news.
Since it is difficult to find representative random-sample polling on all countries, we can supplement this with data from YouGov. A word of caution is in order, however: YouGov does not appear to use random-sample polling, instead relying on online responses weighted by various factors.
This site suggests that 19% of the population likes Bahrain, whereas 13% dislike it. 23% of the population likes Kuwait, and 27% dislikes it. 18% of the population appears to like Oman, while 19% dislikes it. 22% like Qatar, while 17% dislike it. 22% like Saudi Arabia, whereas 44% dislike it. 21% like the UAE, while 25% dislike it. Note substantial differences in absolute percentages, albeit not in qualitative results, relative to recent polling for Saudi Arabia from Gallup. This may illustrate limitations of YouGov's methodology (or less likely, Gallup's) or differences caused by different question framings.
In all cases, millenials and men are more likely to have a positive view of these countries. Unlike in the case of the UK, a breakdown by political affiliation is not provided.
A major overarching trend in polling from YouGov, as mentioned previously, is higher approval for all countries considered among younger generations. One possible reason is that this may represent, not agreement with a country's foreign policy positions or domestic political values, but rather a lower degree of automatic anti-Muslim, anti-Arab, or simply general xenophobic sentiment.
YouGov data suggests that Bahrain is unpopular in the UK, with 20% approval and 23% disapproval. Politics appears to play a substantial role: the Lib Dems are most likely to like Bahrain, with Labour being less likely and the Tories being much less likely. Men are also more likely to like Bahrain, despite being somewhat less likely to vote for the former two parties. Approximately the same percentage of Remainers as Leavers liked Bahrain, despite these positions being heavily correlated with party! It is unclear what this means: perhaps this is indicative of some of the mentioned limitations of YouGov's methodology.
According to YouGov, 17% of people in Britain like Kuwait, whereas 35% dislike it.
YouGov suggests that Oman is currently unpopular in the UK, with 15% liking it and 22% disliking it.
YouGov suggests that Qatar is currently unpopular in the UK, with 17% approval and 34% disapproval. Popularity is higher among millenials. Popularity also seems to be notably higher among people who voted to Remain in the Brexit referendum. The popularity of Qatar seems to have decreased over time.
Data from YouGov suggests that Saudi Arabia is currently unpopular in the United Kingdom, with 47% disliking the country and 17% liking it. There appears to have been little change in these values over the data collection period shown. Popularity is notably higher among millenials, though still low, at 28%.
YouGov suggests that about 24% of Britons like the UAE, while 35% dislike it. Labour is more favorable, while the Conservative Party is less so and the Liberal Democrats are least favorable.
Overall, what does this all mean? In general, in the United States, most people do not seem to have very strong opinions about the GCC countries, with "neutral" usually being a more common response than negative or positive combined. This may be influenced by the generally lower level of recognition of these countries shown in the same polling. The exception is Saudi Arabia, where opinion is clearly negative.
In the United Kingdom, most people with an opinion one way or the other seem to disapprove of all GCC countries, with Saudi Arabia showing the highest disapproval, and significant differences in approval in some cases across political affiliation. The number of people with a positive or negative option tends to be higher than in the USA.