Did Britain look into taking California?

Question:
Did Britain look into taking California?

Short Answer:
Yes, Beyond looking into there were plans conveyed to the British Government by it's Ministers. The United States border with the British in Canada in the far west was disputed until June 15th 1846. Prior to this there were various British plans to assume or compete with the United States for parts of California. Any such colony would strengthen the British claim to the North West.

The various reports of Great Britain's interest in California center around some sort of economic default or exchange with Mexico around the 1840s. A third party transfer of California to the English via British bondholders of Mexican debt or a direct transfer/purchase of California by England from Mexico.

British pursuit of California went so far as theatre government ministers advocating in favor of various plans to high ranking British Ministers in London, but never received favorable responses from London.

Detailed Answer
On August 30, I841, British Minster to Mexico, Sir Richard Packenham, wrote to British Lord Palmerston, 2 time Prime Minister who "dominated British Foreign Policy 1830 to 1865". In this letter the British Minister to Mexico writes in favor of the use of British Bond holders to assume first a British settlement and then a colony in California.

Annual Publications of the Historical Society of Southern California, Volume 12
It is much to be regretted that advantage should not be taken of the arrangement some time since concluded by the Mexican Government with their creditors in Europe, to establish an English population in the magnificent Territory of Upper California..... by all means desirable in a political point of view, that California, once ceasing to belong to Mexico should not fall into the hands of any Power but England. and the present debilltated conditions of mexico, and the gradual increase of foreign population in California render it probable that it's separation from Mexico will be effected at no distant period; in fact there is some reason to believe that daring and adventurous speculators in the United States have already turned their thoughts in that directions......
If it were to be known that an enterprise of this kind would receive the sanction and support of Her Majesty's Government. properly qualified persons would readily be found to carry out the plan; and I am sanguine enough to believe that the result would be the establishment of a prosperous colony united in feeling and interest with England, and at the same time the attainment of an object in my humble opinion, of the highest political importance. I need scarcely observe that any foreign Settlement in California would for some time to come be nominally dependent on the Mexican Republic; but this state of things would not last forever, nor while it did last would it, I image, be attended with serious inconvenience.

• English Interest in the Annexation of California
• Annual Publications of the Historical Society of Southern California, Volume 12

To add to the accepted answer, here is some additional information perhaps indirectly related to the question. This plaque in San Francisco, California Historical Landmark No. 819, includes the phrase "this venture caused wide speculation about British intentions".

Part of the background to this may be this letter (page 1 and page 2) that William G. Rae, the Hudson's Bay Company's Chief Trader, wrote to Juan B. Alvarado, then Governor of California, on 1st November 1841, asking him to issue an order preventing Captain John Sutter from interfering with the Company's trappers and he also asked for a license to enable the Hudson's Bay Company's employees to travel to "whatever part of California the Company's business may require". Wikipedia suggests that Sutter favoured the French taking control of California and Sutter also had ideas of declaring California a republic no doubt with himself as dictator as can be seen from the extracts shown here of a letter he wrote on the 8 November 1841 to Jacob Leese.

The extracts read as follows: ".....The people don't know me yet, but soon they will find out what I am able to do. It is to late now to drive me out the country, the first steps they do against me is that I will make a Declaration of Independence and proclaim California for a Republique independent from Mexico. I am strong now........"

(This letter is referenced in Bancroft's History here and is held in the Vallejo Family Papers Collection reference BANC MSS C-B 441 at The Bancroft Library, University of California, Berkeley.)

Of more direct relevance perhaps to the question were the reports sent to the Foreign Office during the period 1843-1846 by James Alexander Forbes, the British vice-consul to California. His reports are described and discussed in the following article but there is far too much information on British intentions regarding California in this document to attempt to summarise it. Here is just one quote:

"With California in a state of turmoil, an American takeover became more likely. Forbes stated: 'I feel myself in duty bound to use all my influence to prevent this fine country from falling into the hands of any other foreign power than that of England.'"

A British Consular Agent in California: The Reports of James A. Forbes, 1843-1846 by RUSSELL M. POSNER

The final words perhaps on this answer should be those of Sir Robert Peel, British Prime Minister, in his answer in March 1845 to a question in Parliament on the matter of Britain's interest in California:

"I beg to state, in answer to the question of my hon. Friend, in the most explicit manner, that I am not aware of the existence of any such correspondence, and that I believe the report to be as utterly without foundation as any report that was ever invented. I hope that this contradiction may prove a caution to persons out of doors how they put confidence in such stories. I have seen a great many reports as to the undermining ambition of Great Britain; but I have considered that they were propagated rather as a palliation for the conduct of others, than as having any truth in them. I can answer for it that the present Government have had no such correspondence; and as I find no trace of it in any of the offices, I believe that the late Government was equally innocent in the matter. I repeat that the report is utterly destitute of foundation."