I assumed good answers would flood into this one, as it is a fantastic question.

Since i don't agree with some of the answers i thought i'd put my two cents in.

This of-course depends who you mean by Britain. /u/CapnShimmy rightly pointed out that many assumed that the USA would not last very long. David McCullough's John Adams is excellent by the way, i am re-reading it. This belief was not unfounded. Loyalty to the king had a long and rich history of waxing and waning. England even had a go at Republicanism in the 17th century, but sure enough the monarchy was restored.

But more than this there were other reasons to assume the USA would return to the fold. In many ways they had seceded from a position of privilege, as a respected part of an immensely powerful and sophisticated global empire into an autonomous embryonic republic caught between the Appalachians and the Atlantic. Trading was more difficult, McCullough discusses Adam's attempt to secure credit from France, the Netherlands and i think Russia. It is also worth remembering that only 13 of the 30 British American colonies had rebelled, most remained loyal (Bermuda, Canada, and various central and southern American colonies). Even within America's new borders, there were an awful lot of loyalists.

On top of all of this, a lot of the elite of the USA still considered themselves English in many ways, i remember McCullough writing about the relief John Adam's felt in England (post revolution!). He was the American Minister to London, and he felt more at home in a Protestant English speaking country, enemy nation though it may be, than he had done when domiciled in France, America's greatest ally.

There certainly were attempts to undermine american sovereignty. Britain did burn down the White house, we dint stop arming native Americans for instance, just because the treaty of Paris had been signed. What's more is the British Empire and the United States of America as political entities had constantly evolving borders through-out the 19th century. This lead to all sorts of weird and wonderful minor conflicts and border skirmishes, most of which were surprisingly civil.

Someone who certainly did seriously propose reconquest or reunification in the 19th century was Cecil Rhodes. Very interesting man, he had a country name after him, he may have been the wealthiest man in the world and he was certainly one of the most powerful. He served on the Privy Council for quite some time and he was very expansionist.

He wrote "Confession of Faith" in his very early twenties, here is an excerpt (Page 222) that may interest you.

It is easy now, to say that was just a crazy fantasy but his dream to set up a British Colony stretching from Cape Town to Cairo was being realized. To me that seems an even more absurd plan but off he went and for a while you could, in theory, walk from Cairo to Cape Town without even leaving the British Empire. Mr.Rhodes was certainly a man of action, perhaps if he had lived a little longer he may have turned his eye westward.

If your one for conspiracies, you'll love as i do, that he left money in his will:

"To and for the establishment, promotion and development of a Secret Society, the true aim and object whereof shall be for the extension of British rule throughout the world, the perfecting of a system of emigration from the United Kingdom, and of colonisation by British subjects of all lands where the means of livelihood are attainable by energy, labour and enterprise, and especially the occupation by British settlers of the entire Continent of Africa, the Holy Land, the Valley of the Euphrates, the Islands of Cyprus and Candia, the whole of South America, the Islands of the Pacific not heretofore possessed by Great Britain, the whole of the Malay Archipelago, the seaboard of China and Japan,* the ultimate recovery of the United States of America as an integral part of the British Empire, the inauguration of a system of Colonial representation in the Imperial Parliament which may tend to weld together the disjointed members of the Empire and, finally, the foundation of so great a Power as to render wars impossible, and promote the best interests of humanity.**"*

I hope this helped, sorry its long.

Bit of a basic comment here but they did look at reconquering a small part of the States in the War of 1812-15. They invaded the future region of Maine and started fortifying themselves there with the intention of creating a colony called 'New Ireland' should the war proceed favourably. This was during the period when the US-Canada border was very poorly delineated and frequently disputed over, so it's not too surprising that the British held a view of annexing the land. British operations went well and they established themselves in Bangor and other Maine settlements, defeated the Americans in the Battle of Hampden and began setting up an administrative regime.

At roughly the same time (September 1814) the Siege of Baltimore and Battle of Plattsburgh resulted in British checks, which led to the British negotiators at Ghent paring back their demands and the dropping of claims to New Ireland and an independent Native American state on the Ohio. Had Baltimore fell and/or the Plattsburgh offensive resulted in it's ultimate aim of capturing Albany and threatening New York City, it's likely that New Ireland would have been annexed and parts of what is now Maine would have evolved into a Canadian province.

This was only one small part of the States though and even it's status as American territory was in dispute. Aside from this the British tended not to want to reconquer land in North America, they very quickly accepted their defeat after the American Revolution and were more than content to profitably trade without the burden of ruling the Americans and providing defence for them. Every now I then I read passing references to the attack on New Orleans being part of a British plan to conquer the Louisiana Purchase but I've not read enough about that subject to verify it's claim and I've certainly never seen this confirmed in any detail. I believe it's probably more than likely American propaganda seeping into how people write their secondary works. It'd be a rather strange departure from post 1783 British policy towards the United States if it was true.