If nothing else, the UK would more closely resemble central Europe, as I understand it - colder winters, hotter summers
Think about New York, is at 40 degrees N latitude. This means that if NY would be in the Atlantic coast of Europe would fall in Portugal! a little North to Lisbon! (38ºN). Sometimes Is easy to forget that NY is "Southern Europe" league. Southern than the North Coast of Spain.
Now lets see Quebec (53ºN). Well, London is 51ºN. Without the sea current bringing hot tropical water, London should have practically the same climate than Quebec, and 90% of UK is higher than London, so would enjoy the full whole Canadian winter experience.
So if you remove this effect for both Europe and US, NY should be much colder than it is currently (And most of Europe also).
The proximity to the ocean and the general westerly winds of the mid latitudes mean that the west coast gets a moderating marine influence. The cold ocean surface provides us "air conditioning" in summer and yet is also a mild "heater" in winter, helping keep coastal air above freezing temperatures.
But, this general background pattern is regularly disrupted by large high and low pressure systems that wander through the region and give us our weather of storms, cold snaps, and heat waves. These cyclonic systems can both carry a somewhat distinct air mass inside them and also disrupt the normal westerly surface flow. And the jet stream is largely in control of steering these systems, making them track further north or south or even helping them stall out temporarily in their generally eastward course.
The Pacific ocean has a similar current to the Gulf stream (AMOC) called the Kuroshio current. If the Gulf stream stops, you should expect the climate in London to be worse than the climate in the center of northern Canada.
Everything I've read about the Kuroshio current only talks about its effect on Japan. Does it carry all the way to the North American west coast? Looking at this map[0], you can clearly see the effect of the AMOC on how warm the waters north of Europe are compared to similar latitudes elsewhere, but there doesn't seem to be the same relationship between the waters east of the West Coast vs. west of Japan. If anything, the Californian/Oregonian shores appear cooler than the same latitudes in Japan.
[0]: https://www.ospo.noaa.gov/data/sst/contour/global.c.gif
but it's complicated, many other ocean currents exist and affect weather and wont stop
and basically only the overturning part of the AMOC is at risk of stopping, which means e.g. the gulf stream won't stop, but will likely become a bit colder and maybe a bit slower too
Also geography making it more, or less, likely for constant wind from the inland affecting the wetter can also affect the normal expected wetter noticeable.
E.g. UK being a island means there are much less affected by inland wetter moving over to them then many typical Canadian comparison places. (And from-inland wind is the main cause of very hot summer or very cold winter wetter in many places e.g. cold weeks in north Germany, or e.g. the main source of Blizzards in the US).
So IMHO comparisons like "this city is on the same latitude and also on the cost so it will have similar weather" without the overturning current are pointless (which doesn't mean that they can't happen to be correct, but they as much can happen to be fully wrong).