Because the opening post talks about Louisiana, the Oregon Country, Alaska, and Canada (with Rupet's Land being assumed to be included in this) being unified under the British. It does not talk about them then going further south. If you are talking about border revision in the same way the US originally did, in that they got rid of a lot of bumps, making the American-Mexican border three straight lines and two or three rivers, then fine. Otherwise we should not make too many assumptions. Having the Texans and Californians on the British side would be desirable for them, but you are as likely to have American filibusters there.
I'm just proposing what I think would be a logical result. Our POD is 1802. With how empty the Mexican Far North was (even compared to Mexican states like Sonora, etc.), it does seem like that someone might try and force the issue. Britain just needs to have a lobby advocating purchase of those territories, in exchange for both the land and other favours (Mexico could get a lot out of it, let's just say). How would an Adams-Onís Treaty between Spain and Britain work?
A mass annexation of Mexican land (aside from the Mexican Cession lands, US could've grabbed so much more, including
all of Mexico) is much less likely with Britain than the US, but border revisions strike me as plausible. It could just end up a bunch of separate republics. Using the Mexican-American War-era state borders, I could see the most likely borders being a separate Alta California republic, a Texas Republic (minus Coahuila, since it was Coahuila y Tejas after all), and then a Nuevo México republic. Texas and California will be Anglo, Nuevo México Spanish. Alternatively, Nuevo México could stay within Mexico as a massive panhandle. I don't know if Texas and California would remain independent or otherwise join the British Empire. I think we're assuming for this scenario that all of these colonies (Columbia, Rupert's Land, Upper and Lower Canada, Louisiana, etc.) all confederate into one colony to begin with, so I suppose that Texas and California join this confederation regardless of opposing arguments TTL.
And then you have the argument of how to settle this border. Well, the Rio Grande is very long river and a very nice border, but it's up to the diplomats as to where to draw the border from there using unfortunately straight lines. You can't (or ideally shouldn't) use the Rio Grande to its source, because that cuts Nuevo México, the biggest area of Spanish settlement north of the Rio Grande, in half. And the Rio Grande wasn't even the border in the Mexican era. There's a lot of choices to make. Incidentally, because of the reality of borders, you'll still see tons of major cities like Ciudad Juárez/El Paso (Juárez would be a small town named Paso del Norte if you drew the border differently) along the border.