Let's start off with the text of the declaration itself. It is not a lengthy statement:
His Majesty's government view with favour the establishment in Palestine of a national home for the Jewish people, and will use their best endeavours to facilitate the achievement of this object, it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine, or the rights and political status enjoyed by Jews in any other country.
The two key pledges there are:
to facilitate the achievement of ["a national home for the Jewish people" in Palestine]
and to do so in such a way that:
nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine.
The second pledge is easier to evaluate, as despite the good intentions of its inclusion it proved to be flatly contradictory with the first aim. There is no version of some hypothetical Palestine that could have been created wherein British permission of immigration for one set of people as the Mandatory power governing the people residing in that in country who didn't want them there is not a violation of the civil rights of the non-Jewish communities of Palestine. Having a say on who does and does not get to live in your country is generally regarded as being a very basic question of what constitutes the civil rights of a body politic. Or for that matter that simply allowing such a large shift in population (even if it is merely implicit) would not alter the future democratic powers of the people already residing there.
That is of course without getting into ways in which, for example, the British cooperated with organizations leading the Zionist project and the approach of Zionists towards the project itself that we can describe as involving prejudice to the rights of the non-Jewish communities in Palestine. I'm not an expert on those topics but my understanding is that British efforts to defend Zionists necessarily entailed collective restrictions on whole quarters of the non-Jews of Palestine, including in some circumstances mass arrests and detention, disarmament, and collective punishment of villages, for instance by destroying homes.
Evaluating the first pledge is a bit trickier. The inclusion of Balfour Declaration in the text of the British Mandate for Palestine as issued under the league of nations shifted the pledge from merely that, a promise, to a binding legal obligation upon Britain as the mandatory power. That being said, the extent to which they carried that out is up for debate.
Firstly, a "national home" is to my knowledge not a concept that had ever existed in international law (or any other kind of law) prior to the Balfour Declaration. The phrase itself and the context of what that home would be was subject to several revisions before it was eventually issued. You can see the versions of these drafts here. Among other things, the drafts show a shift in language from reconstituting Palestine in toto to being such a national home to the more modest pledge that some such home would merely be established in some unspecified portion of Palestine.
As it is, the phrase is virtually meaningless, disconnected as it is from any promise towards statehood, or of citizenship, or partition or specificity about what portion of Palestine (if any) ought to so function as a national home for the Jewish people. So from a strictly legal perspective it doesn't strike me that the pledge really promises anything other than a guarantee of the rights of Jews in Palestine and implicitly the right of Jews to go to Palestine.
Within the limited scope of that promise however, the most controversial feature of British policy towards Palestine has been the quota system on the number of Jewish migrants that was imposed from the start of the mandate period. Prior to 1939, Jewish migration to Palestine was modest. The vast majority of migrants would head to the United States, but the US virtually ended Jewish immigration with the immigration restriction acts in 1924. While this increased the number of Jews heading to Palestine, the overall figures in third and fourth Aliyah (i.e. migration waves to Israel) from 1919-1928 was modest.
Although Zionists typically advocated for free immigration, ultimately the quotas were negotiated between the British and what would eventually be called the Jewish Agency. Zionist leaders were in principle cognizant that immigration had to be tied to the capacity of the country to receive the new immigrants (i.e. to find them work) and that the immigrants be of value to the Zionist project. In other words, as one migrant wrote, they wanted "young people who know how to wield a hoe and hold a rifle." (as quoted by Tom Segev in One Palestine, Complete. This policy quite explicitly meant not taking in indigent Jews. The Zionists wanted to attract economic migrants, not refugees-- they were in little position to help such people in any event.
The rise of Adolf Hitler and the crisis for European Jewry prior to World War II necessarily shifted the needs and views of Zionists and the wider Jewish community. Unfortunately just at this moment of greatest need what had previously been a cooperative relationship between the Zionists and the British government essentially broke down. The British wanted to shore up Arab support, did not have particularly high view of Zionists in the 1930s, no longer viewed (as they had at the issuing of the Balfour declaration) international and particularly American Jewry as vital for their efforts.
Following a conference between the British government, Palestinian Arabs and Zionists in 1939 it was decided, as part of the White Paper of 1939, the Jewish migration to Palestine would be restricted to 75,000 people over the following five years, and that at the end of 10 years a binational (i.e. non-partitioned) independent state would be established.
Zionist leaders were livid. Chaim Weizmann (who would later be President of Israel) said that the White Paper was essentially a revocation of Balfour. Others compared it to the Nazi Nuremburg laws.
While I think that's certainly a fair assessment of the '39 White Paper, I don't think these immigration restrictions can or should be regarded as having had a substantial impact one way or the other on the overall outcome of the Holocaust. As Tom Segev writes:
The restriction on immigration levied a price in human life, but the White Paper's role in the Holocaust is, in the end, relatively small. In the summer of 1941, Chaim Weizmann estimated that when the war ended it would take twenty years to bring a million and a half Jews to Palestine; Ben-Gurion argued that three million could be brought in ten years. Palestine, then, was not the solution for the Jewish people; the only way to save them and millions of non-Jews was war. Most of the Jews who survived in Europe were in fact saved thanks to the defeat of Nazi Germany....Large question marks hang over several major rescue initiatives [but] in any case, while more Jews might have been rescued, neither the British nor the Zionist movement could have saved millions during the war.
After the war, the British essentially wanted to wash their hands of the issue but became embroiled in fighting with a Jewish insurgency between 1945 and 1948. The UN partition plan, which was the basis for British withdrawal, was not actively supported by the British in any meaningful way, and Britain would ultimately withdraw from Palestine. The British mandate expired at midnight on 14 May at which time Ben-Gurion declared, in the Israeli Declaration of Independence, that it would be succeeded by the new state of Israel. Where the United States and the Soviet Union recognized the new state immediately (the US 11 minutes after midnight, the USSR a few days later), the British did not de jure recognize Israel until 1950, and abstained from its entry vote to the United Nations in 1949.
I think my TL;DR would be that the British kept up their end of the bargain to the Zionists under the Balfour declaration until 1939, at which time there was a substantial shift in British policy and restriction on Jewish immigration to Palestine. This should probably not be understood as British culpability in the Holocaust, however.
The promise that Britain would do nothing to "prejudice the civil and religious rights of existing non-Jewish communities in Palestine" was inherently contradicted by the very aims and policies designed to carry out the declarations promise of "the establishment in Palestine of a national home fore the Jewish people", and can therefore be regarded as a flatly broken (and, indeed, untenable) promise.
Source wise:
As I quoted from repeatedly, I quite like Tom Segev's One Palestine, Complete.