Undoubtedly the book Krazer suggested would make for the most thorough answer, but the Japanese wikipedia article on the 国語審議会 (the Japanese Language Council) has got some interesting details.
From 1949 to 1961 the chairman was 土岐善麿, a supporter of the switch to romaji. Some of his work was published in romaji. (At least one example, Nakiwarai, is online if you'd like to see some period romaji text). Members of the カナモジカイ, a group promoting the use of kana only, were also involved. There was a sub-committee for dealing with issues around romaji, ローマ字調査分科審議会, from 1950 to 1962. This tackled issues like determining a standard for romaji, and the use of romaji in education.
For example, here (1957 - official report) they report the improvement of results (referencing Japanese and maths classes), although they did want to do a greater number of tests. It also talks about romaji use in society in general - an interesting example given is that for punched tape, it took the least space to encode romaji compared to kana or kanji. The new technology of the time was not particularly kanji-friendly.
At this point it seems there was still the intention to move towards script reform. However, conflicts between those in favour of (改革派) and against (慎重派) reform came to a head in 1961, when five of the 慎重派 walked out (see final remarks by 成瀬正勝 at the end of this page) over arguments about how elections were handled. Specifically, I think the feeling was that the 国語審議会 had been formed mostly from reformists under the control/influence of the occupying forces, and that the same people were being constantly re-elected.
The Ministry of Education got involved, asking for a re-evaluation of reform plans. One of the outcomes of this was that the ローマ字調査分科審議会 was dissolved. 土岐善麿 was not only no longer chairman after this, he seems to have been removed entirely. This, from the next meeting after the walkouts, summarises some of the viewpoints that had come up critiquing the work of the council, including suggestions that the influence of the romaji and kanamoji groups had complicated things, and that the proposals of the council should not be forced on people.
The final nail in the coffin seems to have come out of a 1966 meeting where the address given by the Minister for Education, 中村梅吉, included these lines stating that the use of mixed kana/kanji was pretty much a given:
After this, the 国語審議会 focused more on subjects such as the daily-use kanji lists. In 1968 the amount of time given to teaching romaji in 小・中学校 was also reduced.