So by the time the “Holy Lance” was discovered, the Crusaders had already taken Antioch and were defending it against a relief army under Kerbogha, the Atabeg of Mosul. If the sally by the Crusader army against Kerbogha fails for whatever reason (lance or no), the result is the likely extirpation of the Crusaders.
It should be noted that some of the Crusaders have already gone elsewhere - specifically, Baldwin of Bologne’s departure for Edessa. Baldwin seized power in Edessa in March 1098, while the Battle of Antioch occurred in late June. Thus, ITTL the “County of Edessa” is still created, although as the only Crusader state it may not be very long-lived.
The immediate result of a Crusader defeat at Antioch is going to be the aggrandizement of Kerbogha, who will become ruler of a state stretching from Mosul to the Mediterranean. Edessa may be his next target. But Kerbogha’s position is unstable - his army was a fractious coalition whose leaders were probably more worried about Kerbogha’s own power than some Franks capturing Antioch. While taking Antioch might strengthen Kerbogha, it might also prove to be an overextension. I suspect sooner or later his power is going to crumble due to internal unrest and external threats. Duqaq and Ridwan, the brothers controlling Damascus and Aleppo (respectively), are bound to oppose him, and the Fatimids are enjoying something of a recovery to the south, having taken Jerusalem back from the Turks around a year before the Crusaders captured the city IOTL.
Certainly Kerbogha will not be much of a threat to Alexios; he’s got enemies much closer to home to worry about. From the emperor’s perspective this whole Crusade thing has gone rather well - with the help of the Franks, he’s managed to liberate Nicaea and other parts of western Anatolia, and their defeat at Antioch doesn’t really harm the Byzantine position. The Sultanate of Rum will remain a threat, but there’s no reason that they would suddenly overrun the lands that Alexios has reclaimed because of what happened in Antioch. Indeed, it’s entirely possible that the defeat at Antioch will be good for the Byzantines. Alexios may still be able to assert control of Cilicia after the defeat of the Crusaders, won’t have to worry about Bohemond’s attacks on his territories, and if (or more likely, when) Kerbogha’s state collapses the Byzantines may well be able to pick off Antioch for themselves. In the long run the failure of the First Crusade undoubtedly butterflies the disaster of the Fourth, so this could be a pretty good TL for Byzantine fortunes in general, although that depends at least as much on Byzantine succession as anything the Latins are up to.
Perhaps more interesting but far harder to measure are the effects this would have on the Latin West. Does this failure redound badly on the Pope and the nascent “Crusading movement” in general, and if so, how badly? I can’t see the urge for holy war being snuffed out by a single defeat, but clearly there will be consequences. Perhaps the notion of the holy war being prosecuted by mere noblemen is discredited and the Papacy more strongly urges kings to take the wheel as they later did - most famously in the Second Crusade, but for the first time in the “Norwegian Crusade” of 1107-1101. Or perhaps Christian militant fervor will be redirected elsewhere, to more accessible targets like the Muslims in Spain and the Pagans in the northeast.
