Well, that’s a little different - we were discussing USA vs. UK (the British Isles). Throwing in hypothetical allies does change things a bit.
Mexico used to own Texas, Nevada, New Mexico, Arizona, California, and I think maybe Colorado and on up the west coast the Canadian border. Yeah, they’d probably like to have all that back. But, I can’t recall them ever defeating the US in a war. Yes, they did win at the Alamo - but at the time Texas was NOT part of the United States. So, while they might have wanted to join in, by 1913 they were getting this feeling that attacking the US maybe wasn’t such a good idea, because every time they did they lost territory. (I think Mexico is still a little peeved that we gave Japan back to the Japanese in the 1940’s, but they haven’t gotten any of their territories back)
Other potential allies? Other than Canada, none with a land border with the US.
Mexico attempted this several times - remember, they occupied the US Southwest and were forced out, and made more than one attempt to get the territory back. Never managed to do so. They answer is no, backed up by history.
Again, I’m asking you just how much damage do you think the RN could actually do in 1913. Blockade? Fine, we’ll trade more with Asia. No blockade on the Atlantic coast alone could possibly qualify as a “stranglehold”. Inconvenient, yes. Annoying, yes. But it doesn’t stop west coast trade, or overland routes (Mexico might well ally with the US rather than Britain to appease their northern neighbor). Plus, what did the US lack up until 1913 that they were forced to go to Europe or Africa for? Indeed, the rest of Europe might have protested at being unable to buy our raw materials and oil - we were the top petroleum producer at the time, if I recall. Definitely we were self-sufficient food-wise.
So the hypothetical blockade must do more than blockade - it must also attack. Which cities? How many ships apiece? At what cost to make us surrender? May I remind you that a failed British navel bombardment, at a time when the the RN truly outclassed the infant US fleet, was the inspiration for our national anthem? Perhaps I should also remind you that Americans display a surprising tolerance for blood baths when they believe in the cause they’re fighting for - WWII never touched US soil after Pearl Harbor, yet we laid down tens of thousands of American lives for the cause.
Also, all that cleverness that in peacetime results in such innovations as the assemblyline, aviation, computers, TVs, Britney Spears CDs (OK, maybe that was a bad example) and other fun consumer goods gets turned to more sinister purposes during wartime. The UK would also have to deal with the inventiveness and viciousness of people who have given the world such items of warfare as the machine gun, landmine, submarine, and trench warfare - which I will also point out that we invented prior to 1913. In fact, we came up with them way back in the 1860’s and used them on each other. We’re extremely nasty in a fight.
You can’t neglect the psychological aspects of a battle or a war. In the type of seige you’re proposing, victory comes when the other side gives up. What are you proposing the make the US give up in this sceanario? A bombardment? Did that induce the UK to surrender in WWII? Or did it make them more determined to resist? Americans have no less affection for their territory than the British do. With an entire continent worth of resources to support the country, how will you “starve” them out? Even if we had to abandon some of the east coast cities, would that actually make us surrender, or accept British terms? Unless you’re going to stage a land invasion to finish the deal I don’t think this would work (and we’ve pretty much agreed a land invasion wouldn’t work either). All the US would have to do is hold out until the British get tired and go home. It’s an effective strategy. It’s what won the war in Viet Nam for the Vietminese, it’s what worked against the USSR in Afganistan. It’s one instance in which a small, third-rate country can take on a superpower and win. Between 1865 and 1913, the US was not a superpower. but it was not on the bottom of the heap, either.
But, by all means, please tell me how you think an east coast RN bombardment would bring about the surrender of the US and agreement to terms.