The Bank of England was nationalised, in 1946! I assume you mean why the wider banking sector wasn't nationalised as well, though?
/u/AlviseFalier is certainly right to ask the question of: "How would the Labour Party nationalise the banking sector?". I want to expand on this by asking: "Why would the Labour Party nationalise the wider banking sector?" In the end, Labour's commitment to nationalisation was not as unwavering as some may believe, and its measures were limited by pragmatism, especially in the political sense, as well as changes in economic theory and practices.
In order to understand the extent of Labour's nationalisation programme, it may be helpful to start by looking at the nature of Labour's socialism. Let Us Face the Future, Labour's election manifesto in 1945, was undoubtedly a socialist manifesto. But what did the Attlee government mean by socialism? Francis has argued that Labour's definition (from an analysis of the 1949 policy statement Labour Believes in Britain) was:
"a commitment to the pursuit of equality, both of opportunity and of property, the transfer of economic power through public ownership, and the maintenance of full employment and productivity through physical planning" (Francis, p.222).
The relevant part for this discussion is the "transfer of economic power through public ownership", of course. Yet Labour's socialism was not just about the material improvement of the masses and economic efficiency. It also involved a strong ethical line, which strongly influenced policy-making and defined Labour's nationalisation problem also, which I will develop on.
Labour's post-war nationalisation programme undoubtedly went above and beyond the expectations raised during the wartime period. The 1945 manifesto had promised a series of nationalisation waves, reflected in the government's policy:
Nationalisation examples include (Thorpe, p.123):
1946: Civil Aviation and the Bank of England
1947: Coal, rail, road haulage, and cable and wireless
1948: Electricity and gas
But after 1948, for the most part, Labour's enthusiasm for nationalisation (which may well have included further nationalisation of the banking sector) fell away. Far fewer industries and services were earmarked for nationalisation. Some that were, such as the water and cement industries, were giving far less priority. Why?
Certainly, there were concerns over the popularity of nationalisation with the electorate; after all, ultimately Labour's job was to win elections. In 1945, 60% of the electorate favoured the nationalisation of coal (Fielding, p.634). But by 1949, ministers were less certain. The nationalisation of the steel industry was heavily debated, and although passed, there were nonetheless serious concerns over its popularity. This ties in with the Labour government following Herbert Morrison's line of "consolidation"; that further nationalisation should be held off to evaluate the success of recently nationalised industries. Fielding has also highlighted the influence of Labour's ethical socialist side here. It appeared, to Labour leaders, that legislation was going beyond the expectations of the people and Labour should use "the opportunity to expand its programme of socialist education” (Fielding, p.151). The political theorist and member of the Fabian Society G. D. H. Cole supported this, believing that “the making of more socialists was the necessary precondition for further reform”. So further nationalisation was held off for the time being, even if Labour ministers such as Attlee and Dalton were still committed to increasing public ownership in the future (Francis, p.151).
This is only part of the story, though. There was no brief pause in order to consolidate; Labour seemed to hold off from further nationalisation indefinitely. In fact, in their 1951 manifesto, there was not any explicit mention of nationalisation! Clearly, something more must have happened for Labour to move away from one of its 1945 foundations of its socialism. And that was the move towards the adoption of Keynesian economic techniques.
Now I must stress that I am not an economic historian, and my understanding of Keynesian economics is framed in political terms. I'm sure you're far more knowledgable in this area, so the next bit might not be needed for your understanding. But I'll spell it out anyway in case anyone else wants to read this (and I apologise if my terminology is a tad vague). To drastically simplify the situation: Classical economic theory argued that supply and demand always met and balanced out in equilibrium. Keynes claimed this to not be true, and that instances of high unemployment occurred when the economy experienced too much supply and not enough demand. The most significant part, though, was his argument that consumer demand could be increased by raising government expenditure and/or by cutting taxes.
This is massively relevant because Keynes effectively demonstrated that the government could use macroeconomic policies to manage the economy! The consequence of this was that micromanagement was not necessary: that controls on the economy and the public ownership of businesses were not the only method of economic management. This was, to many traditional socialists, heretical. To the left of the Labour party, Keynesian techniques were a retreat from socialism, their conception being along similar lines of the Soviet model. After all, Keynes himself was a Liberal, and his economics were too. But it was certainly compatible with many Labour supporters. Many Fabian supporters of top-down socialism, for example, found Keynesianism greatly attractive. Hence in the 1940s Labour economists such as Gaitskell had been converted, as well as other intellectuals (such as Jay and Durbin). These Labour members rallied around Keynesian economic techniques, which did not require public ownership to manage the economy. This was not a retreat from state interventionism, but instead entirely different methods of achieving this. The key point, though, is that Labour's retreat from nationalisation was not simply due to concerns over the its appeal to the electorate. New economic techniques, as demonstrated, made nationalisation effectively redundant in achieving Labour's socialist goals.
To bring this all together: Labour didn't commit to further nationalisation of the banking sector because, in this economic context, it was not necessary in achieving Labour's socialist goals. Keynesian techniques rendered further public ownership of the banking sector superfluous. Hence, with considerations over the popularity of nationalisation already influencing Labour opinions, a retreat from further nationalisation seems understandable. Why then, would Labour have any need to nationalise the wider banking sector?
I hope that answers your question, please feel free to ask for clarifications or any follow-ups.
Sources used:
Fielding, S.: ‘Labourism in the 1940s’, 20th Century British History (1992).
Fielding, S.: ‘What Did 'The People' Want?: The Meaning of the 1945 General Election’, Historical Journal (1992). Great for looking at the popularity of Labour's socialism. It shows that the British working class was not demanding socialism; that people wanted life to be easier than before the war, but did not wish to build a "New Jerusalem".
Francis, M.: ‘Economics and ethics: the nature of Labour’s socialism, 1945-1951’ 20th Century British History, 6 (1995). Reconciles Fabian and ethical socialism, demonstrating that they were not so much in contention in this period.
Thorpe, A.: A History of the British Labour Party (1997). An excellent overview for any part of Labour in the twentieth century.
As difficult as it is to prove why something didn't happen, in this case I can answer your question with another question: how would they? With what money would they redeem the shareholders? Sure, the government could simply seize the banks, but that would alarm the United States; the bank's primary creditors.
The Atlee's Labour Party ran on a policy promising an expanded Social Welfare system, and that's what they dedicated themselves to; and it proved enough of a challenge by itself. Beyond that, Atlee's continued policy of maintaining Great Britain as an imperialist power meant that a well-connected private bankings sector was vital to keeping credit lines to the United States open. Indeed, in his book titled European Integration, Mark S. Gilbert illustrates how British insistence on maintaining a military presence on the European Continent, in Africa, and in Asia leads to Great Britain not only to suspending Pound convertibility to the dollar in 1947 (effectively going into default) but also missing out on the entire european integration process: just to give an example, the Treaty of Paris is item number five on Atlee's cabinet agenda in April 1951 because previous four items have to do with communist militias in Indonesia which the British army can't hope to defeat but insist on trying anyway. Hardly the conditions to add, "Banking Nationalization" to the agenda!