Well, the way the primary sources (chiefly William of Poitiers, who was formerly a knight and later chaplain to William I, and was therefore in a position to get his information from those were there) describe it it was up in the air practically until the very end.
It's often speculated that Harold marched south before collecting his army because he wanted to duplicate the success of surprise at Stamford Bridge; William of Jumieges stated that he planned on a night attack. In any case his army had far fewer fyrdmen than it would have if it he had waited a while, but it did have the housecarles from the households of Harold, and his brothers earl Gyrth and Leofwine. If Harold did have to form up his army unexpectedly, he picked a reasonably good place for it- it wasn't exactly on a cliff but it does seem to have been anchored by slopes, and a creek and forest on either side. As far as new English contingents arriving goes, well it can't be ruled out but it certainly doesn't accord with the contemporary descriptions. Florence of Worcester said that many English left during the day. One of the Anglo-Saxon Chronicles said that Harold "fought with the men who would stand with him", which suggests that there was not universal determination.
In any case, the initial Norman attack was ineffective. Their archers (Jumieges also refers to slingers, and the Carmen de Hastingae, though now considered rather dubious, mentions crossbows) didn't make much have much effect against the English shield wall, so the Norman infantry advanced and were met, so William of Poitiers said, with a very wide and very nasty assortment of things thrown by the English. Anyway the Norman infantry had no luck either so the Norman cavalry waded in awkwardly with their swords but also made no impression. Eventually upon a rumour that William had been killed the Normans seem to have routed, and were pursued by at least some English. Historians disagree on exactly what the English did, the greater part seem to think that this was an impetuous action by a minority in Harold's army and that it contradicted a supposed defensive plan- as long as Harold stopped William advancing down the Senlac Road William would eventually have to leave. On the other hand some think Harold ordered a pursuit which was thwarted by the inopportune deaths of his lieutenants Gyrth and Leofwine (their deaths in the Bayeux Tapestry seem to correspond to this point in time).
In any case, when William rallied his forces his cavalry caught those English, who had broken from their shield wall, in the open and disarrayed, and charged them down. Apparently the Normans thought this was a good tactic and spent many hours whittling the English shield wall down by variously riding up and chucking lances at it, drawing some of the English out and charging them down, interspersed with attacks by the Norman missile troops. However, according to Poitiers, even at the end of the day the English were still quite numerous and formidable. Evidently putting it all on the line William ordered his whole force to attack. The fighting seems to have been quite hard and the way that Poitiers and other sources (the Anglo-Saxon Chronicles, the Bayeux Tapestry) describe or depict it, the remaining English seem to have kept fighting until, one way or another, Harold was killed, at which point the army largely dissolved. The Bayeux Tapestry's depiction of this juncture shows the Norman cavalry charging while in the lower margin the archers fire at a 45 degree angle. This, and some much later medieval poetry is the basis for the idea that the shield wall was wrecked by the Norman archers. I guess that is possible. Personally I think it was at least as important that by the end of the day the Norman infantry, who had spent the whole of it sitting around, would have been well rested while the English, who had been fighting continuously, were very tired. There is additional speculation that the casualties among the English by the afternoon's skirmishing were concentrated in one part of the line, making it liable to break and open up a gap in the shield wall. This is altogether possible but not certain.
"That William, realising that the battle could only be decided by either his or Harolds death created a hit squad of knights. It is recorded that they cut him through the heart and head and such, but also cut of his leg. Leg being a polite medieval replacement for the male genitals. I tend to accept this version."
To be honest although it's possible it's based on sources much later than the battle (the poetry of Maistre de Wace I believe) and in historiographical terms was greatly played up by nationalist Whigs like E A Freeman, for whom it represented the ultimate in Frankish deviousness.
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