Edit: I do Early Medieval Chinese History, focusing on political religion (particularly Buddhism), and it's a time when a lot of sarira is coming from the west into China.

after the death of the Buddha, there was a great war that was fought over his cremated remains.

Indeed, this is from Buddhist histories (see John Strong's Relics of the Buddha, Princeton, 2007) and sacred sutras and texts. I quote from Strong's work (in turn citing Buddhaghosa's Commentary on the Digha Nikaya), picking off of where the famous King Ajatasatru attempts to secure the Buddha's relics by military force from the Malla clan, who were lords of Kusinagara, where the Buddha entered nirvana:

Suspecting, however, that his request might well be turned down, [Ajatasatru] backs it up by equipping a fourfold army and setting out himself to Kusinagari.

Much the same sequence of the emotions and events is recounted in most canonical accounts of the parinirvana... In addition to Ajatasatru, no fewer than six other kings come into the contention. The lists of these vary slightly in names and order, but in general, there is agreement that the peoples and countries involved, beside the Mallas of Kusinagari and Ajatasatru of Magadha, are as follows: the Licchavis of Vaisali, the Sakyas of Kapilavastu [note: the Buddha's clan], the Mallas of Pava, the Bulakas of Calakalpa, the Kraudyas of Ramagrama and the brahmins of Visnudvipa... (118)

The stories generally end with the rulers deciding to peacefully divide the relics into eight parts. Cave 148 at Dunhuang, often called the Nirvana Cave depicts this scene in vivid detail (more images are available elsewhere). The Mahayana Mahaparanirvana Sutra (trans. Yamamoto, 2007) even casts the division as something that was ordained by the Buddha himself before his nirvana:

Original: 善男子。如來今於拘尸那城娑羅雙樹間。示現倚臥師子之床欲入涅槃。令諸未得阿羅漢果眾弟子等及諸力士生大憂苦。亦令天人阿修羅乾闥婆迦樓羅緊那羅摩睺羅伽等大設供養。欲使諸人以千端疊纏裹其身。七寶為棺盛滿香油。積諸香木以火焚之。唯除二端不可得燒。一者儭身。二最在外。為諸眾生分散舍利以為八分。

"O good man! The Tathagata now reclines at Kusinagara between the twin sal trees, in the posture of a lion, and shows himself as entering Nirvana, and all those disciples of his who have not yet attained arhatship and all Mallas are smitten with great apprehension and sorrow. Also, devas, humans, asuras, gandharvas, garudas, kimnaras, mahoragas, and others make great offerings. All people are now ready to burn him with 1,000 “tans” [one “tan” = either 16, 20, or 60 feet] of finely woven woolen cloth, in which to wrap his body and lay him in a coffin of seven gems, embalmed with fragrant oil and covered over with all kinds of fragrant wood. Only two tans cannot be burnt off. One is that which is closest to the body, and the other is what is outermost. Among all the people, his remains are to be divided into eight parts...

Is it possible that some of these relics of the Buddha survived to be venerated to the present day? And if so, what is the likelihood that the Buddhist pilgrimage sites that claim to hold relics of the Buddha actually do?

This is a much harder question to answer. I'm far more familiar with sarira in the East Asian context, since that's my field, but the short answer is few people actually even see the relics (see also Embodying the Dharma: Buddhist Relic Veneration in Asia by David Germano and Kevin Trainor), or handle them closely, so it's hard to even do a visual analysis of the object so no one really can say for sure. Certainly, even the very investigation of the authenticity like the relic and Kandy has been very controversial (Locations of Buddhism: Colonialism and Modernity in Sri Lanka by Anne Blackburn, 182) and led to protests in the press. Some European skeptics also claimed that sharira were other things... in the first journal of the Journal of the Peking Oriental Society J. Rhein complained that a relic of the Buddha at Tushita Monastery 兜率寺 was an elephant tooth (180) and also noted the resistance towards authenticating the relic there in 1904.

On a personal anecdote, the temple that I belong to also has the Buddha's relics enshrined in a pagoda and I've seen them... they look like bright yellowish transparent spheres. Ultimately, I think the significance sarira have to Buddhists (incl. myself) is far more symbolic: it's not necessarily mandatory to have something to have been part of the Buddha's body, but a relic's presence as a reminder of his time in the world is very important.

Of interest: relics claimed to be those of the historical Buddha, some of his legendary disciples, and other great and historically famous masters have been collected and displayed around the world in the Maitreya Loving Kindness Tour