Invasion and occupation isn't a really issue. The British have two fully worked up 51's off the western coast of north america - either of which outguns the entire active US Pacific squadron twice. SF will fall as soon as the RN can gather itself together to strike it.
Do you have any examples of the Royal Navy seizing and holding a port without the support of the British Army?
"In the present state of the defences of this harbour [San Francisco] one-half of this force [The British Squadron at Esquimault] could command the city of San Francisco and take possession of this yard [Mare Island]." – Flag Officer Charles Bell
Actually, Bell probably meant the Pacific Station in Valaparaiso. Esquimalt only became an alternate station in 1865. Until 1887, the Esquimault Royal Naval Dockyard lacked the facilities to repair damaged hulls and had to be sent to American dockyards in Seattle for repair.
I'd be interested if Rear Admiral Thomas Maitland also thought capturing the yard in San Francisco would be easy.
With SF gone California is effectively severed from the Union, and the Union will suffer a massive economic depression
You are aware that San Francisco was not the only port in California? And that it was contiguous with the rest of the Union?
(the loss of one California Treasureship in, ISTR, 1860 caused half the businesses in NYC to go under).
I’d be interested in seeing your source.
The state of California is almost defenceless once SF is gone and has absolutely no possibility of arming itself. No help is going to be forthcoming from the "mainland" as there is no capability of it reaching California.
So California is an island in your time line? In ours, it’s contiguous with the rest of the Union. It also had troops, arms, and a population of nearly 400,000.
The loss of SF also undermines the US position in NM etc. - Sibley will likely be successful and "liberate" New Mexico territory.
Even with your rather optimistic timeline, Sibley will have lost the New Mexico Campaign before the Royal Navy appears off the coast of California.
Getting a Corps of British troops to SF for further operations takes considerable time. Orders to sail can be actioned rapidly out of London (who have a cable connection with India), but it's a good 6 weeks sailing with layover
Orders can be actioned rapidly out of London, but they won’t. Multiple elements of the British government are going to have to decide if they send an expeditionary force from India, what size it will be, and whether it will reinforce western Canada, invade California, or attempt both, and who will command. Once that is decided, which could easily take weeks, orders can be sent.
Transport will be needed for the expeditionary force, which will take time to assemble. Stores and munitions will need to be assembled, stored, and then loaded. That adds more weeks and even a rumor of the Indians getting restless will probably abort the expedition. Travel will be at the rate of the slowest transport or supply ship and both they and the gunboats will be in significant danger in a storm.
That gets them to Valaparaiso, where the needed naval elements may or may not be in port. If any cavalry was sent, they’re going to need time to recouperate from the sea voyage as well. Then, once the naval element is brought up to speed, they can begin the joint operation, with all the chances for friction that entails.
Meanwhile California, which is in telegraph communication with Washington DC, will have had several months to raise and train troops, as well as improve and expand fortifications. They may waste that time under the not unreasonable assumption that Britain wouldn't try to conquer California with 20,000 men.