And If so, what was the degree of popular acceptance they achieved ?
/u/gingerkid1234 covered the Christian side, I'll add a little bit to the Islamic side. Fair warning: This is very long, but hey. There's a lot to talk about.
Now, I assume you're inquiring more generally about liberal interpretations of the religion, correct? Obviously the dynamic between Islam and Judaism is quite political right now, so I hesitate to frame it as taking something from Reform Judaism. Instead I'll just talk a little bit about the more liberal schools of Islamic thought. Hopefully this is what you're after; if not, at least you'll know a little bit more about Islam.
With that out of the way, the first thing to realize is that Islam has been going through a conservative revival for a while now. It has roots in the rise of nationalism in the 1800's (for example, the Muslim Brotherhood was founded in 1928), but didn't really get into high gear until the 1970s. The story is pretty well known: oil prices started to rise considerably and production elsewhere (particularly the US) started to decline, giving OAPEC considerable economic power - particularly Saudi Arabia. Saudi Arabia has, since the time of the First Saudi State (1700s), been Wahhabi / Salafi (some consider "Wahhabi" derogatory, but there are subtle but distinct doctrinal differences) - a conservative, quite frankly aggressive and bellicose interpretation of Islam. I could go into greater detail about why Muhammad Ibn Saud (not the later, more well known Ibn Saud of the Third Saudi State) chose to support Muhammad ibn Abd al-Wahhab, but that's not particularly pertinent right now. If you want to know, just ask. Either way, it started as a very fringe interpretation of Islam. With the aforementioned increase in money from oil exportation, the Saudis began to print and distribute Qurans free of charge across the globe, along with specifically Wahhabi literature. This not only led to an increase in general religious interest (popularity of the Hajj has skyrocketed), but was effective at spreading Wahhabi ideology specifically. This had the effect, combined with Westernization in Turkey and Iran, of drowning out the pre-existing more liberal schools of Islam that still existed.
I mention all this as groundwork, but it's important to understand this as something of a watershed moment between older religious thought and newer. The Revival has had (and is having) considerable effect amongst Islamic countries - it's been a game changer and Islam is changing.
As for the pre-revival period, the biggest school of what might be considered "Liberal" Islam is Sufism. Sufism was more popular than it is now before the Islamic Revival (particularly in Turkey before Atatürk closed down the tariqas and in Iran before Isma'il I Safavid) though it's beginning to recover. It's still popular in North Africa as well. Sufism is frequently described as the "mystical" branch of Islam. Sufis believe you can come to know God intimately through striving to reach fitra - essentially meaning purity of spirit. It is a state in which nothing you do is sinful (which is an extremely important concept, which I'll get to in a moment), and everything you do is motivated by love for God. In Islamic tradition, the fourth Caliph, Ali, is regarded as the prototypical (and original) Sufi by way of his well known interest in learning and philosophy, as well as his deep compassion.
Many Sufis believe reaching that state is impossible through logic alone, and so they appeal to the emotional, with a particular emphasis on poetry. This is why Sufism has a longstanding association with poetry, and why many of the great Islamic poets (Jalaluddin Rumi, Ibn Arabi, Al-Ghazali, perhaps Omar Khayyam) were Sufi. They use analogy to express a love of God, frequently employing wine as a symbol for God, and drunkenness as a symbol for the kind of unconditional love where one does foolish things.
Because of this focus inward however, ritual tends to be observed less amongst the lay people, and what is observed tends to change. This led to accusations of Sufis not being true Muslims, friction between Sufis and other schools, and even persecution. Whirling, a well known activity mostly (if not entirely) specific to Sufis, has been a source of contention. Many mainstream theologians consider it Bid'ah - heretical innovation, doctoring the Quran. They also have a history of respecting / venerating the graves of Sufi saints, which was deeply concerning to more conservative Muslims, who considered it idolatry, a very very serious sin. In addition, some Sufis believed that after one reached fitra, anything you did was without sin regardless of what it may have been, because it was by definition motivated by love of God. This opens the door to further innovation as well as a denial of the authority of the ruling class and the imams. This person who reached fitra knew God and was already pure, what use do they have for Imams or the rules of men? This was all particularly important during the Revival because of Wahhabism's more restrictive definition of Muslim, so persecution increased. Attacks on Sufi mosques and tariqas today are not uncommon.
In addition to Sufism, another old form of "liberal" Islamic thought is Mu'tazila, common particularly in Iraq and the Levant. Developed before Sufism, they emphasized rational thought and logic and believed that one could rationally discern the will of God through the Quran and Hadith. When you think of the Islamic study of and commentary on Aristotelian and Platonic thought, that's the Mu'tazila you're thinking about. The Abbasid Caliph Al-Ma'mun was a militant Mu'tazila, and initiated the Mihna, an inquisition where he, and later Caliphs punished those who refused to accept the idea that the Quran was created. This had the opposite effect he intended, and severely upset the otherwise unconcerned masses. This lead to a general distrust of Mu'tazila, and the movement gradually died out. These days, accusations of being a Mu'tazila are used as political and religious insults in Muslim countries, sort of like "Communist" was during the red scare.
As for the modern "liberal" interpretations, there has been a small but growing movement in Islam that emphasizes Ijtihad - independent interpretation - instead of taqlid - imitation - in religious, instead of strictly legal issues. It began late in the 19th century (it's hard to say exactly when, it was a gradual process) as a response to gray areas coming from Islamic contact with Western democracies. It's worthy of its own discussion honestly, but I'll keep it at that. There is serious dispute between Islamist groups about whether or not ijtihad leads to innovation or not. This has parallels with older Mu'tazila thought, though you'd still be hard pressed to find a Muslim jurist or theologian who self-identifies as Mu'tazila.
The 19th century also saw the beginning (or resurgence, depending on who one listens to) of Quranism - the believe that the Quran is the only source of guidance for true Muslims, and denial of the Hadiths completely. They cite passages in the Quran like this one, and point to the initial unwillingness of Caliphs to compile any written Hadith, as according to Ali Nasiri's book An Introduction to Hadith: History and Sources the prophet Muhammad is said to have specifically told them not to write down anything other than the Quran, as it was complete in itself, and required nothing added to it.
There are other more or less "liberal" movements within Islam, some more recent than others, some more liberal than others. But because most of them are so embryonic right now and so disorganized (and frequently argue more specific points, like feminist movements), it's hard to get into much detail.
Now, sources. I'm not sure exactly where you can find each point I bring up, but The history of Islamic theology from Muhammad to the present by Tilman Nagel, and Understanding Islamic sciences : philosophy, theology, mysticism, morality, jurisprudence by Mutahhari Murtaza are both good references. I've also had Sufi friends and spoken with other local (American) Muslims on their beliefs over the years, most of which were liberal. If you're interested in learning more about the modern movements, I would recommend visiting Islamic student organizations at local universities or seeing if you can visit some local mosques. Most but not all schools of thought allow non-muslims into Mosques provided they show respect and aren't disruptive (or polytheists).
I would argue that a Christian analog to Reform Judaism is not really possible. There's a lot of historical and religious baggage that makes Reform Judaism what it is, and most of that isn't shared with Christianity. Even though certain parts might have analogous pieces, there's nothing that could really mirror anything more than a part. It's also important to note that Reform Judaism has changed a ton over time, and so nailing down an equivalent changes with time.
To examine what exactly about Reform Judaism makes it not really equivalent-able, I think it's best to look at Reform Judaism's platforms. These are basically broad consensus statements of American Reform Judaism at the time.
The first is the Pittsburgh Platform of 1885. It's here. While Reform Judaism has lots of antecedents, mostly in 19th century Germany, this is really the first articulated statement of belief by what would become a reform institution (rather than an individual reformer). It has a few basic points:
So, what's the equivalent of that? The concept of an important Christian history to be understood and valued doesn't happen the same way in Christianity. While Christian denominations have reformed beliefs, the focus on specifically accepting or rejecting traditional ritual isn't a focal debate in Christianity as it is in Judaism. Christianity's never been seen as one nation, really. Actually affirming that Judaism is a set of beliefs and not a nation is something that really doesn't make sense as a Christian position at all. The last two could have equivalents, I suppose.
But a later platform, from 1976, has some stark differences. Namely, Judaism is now seen as a people and a religion, Zionism is affirmed (rather than rejected), and ritual is still not required but is an important part of being Jewish.
So, the main points of Reform Judaism are really not things that could have Christian equivalents. Whether or not Christianity is a nation or people or ethnicity has never been a question. While Christianity does have rituals and rules, the sort of legal system Judaism, with a massive centrality of importance, doesn't exist in Christianity, so a religious movement to reject it really wouldn't make sense.
Now that I've written this, I realize I've totally neglected Islam. While a movement rejecting the obligatation of Muslim religious law might make sense in that context, I'm unaware of one existing (hopefully an expert on this specifically can chime in). And the concepts of rejecting or accepting Jewish people-hood is also not a question that really works in Islam, either.
so this seems somewhat answered so i want to pop in and see if people see a connection between this Jewish move and Hinduism.