I. The Battles
In our alternate scenario, the primary change to history is that Harold Hardrada bests Godwinson and the Saxons.  Harold Hardrada was the most mighty Viking leader of his day, renowned as both a warrior in his own right and as a war leader.  In our alternate scenario, let us assume his losses at Stamford Bridge were not cripplingly severe, leaving most of his 300 boatloads of men intact and battleworthy.  Given Hardrada's reputation and record, this is not too great a concession.

Upon defeating Godwinson, Hardrada could have insisted on oaths of service from the defeated Saxon warriors, integrating them into his forces and further swelling his army.  He might also have delayed longer than Godwinson did in reality in order to recruit more Anglo-Danes.

Upon landing in England, William had begun to construct a crude castle backing onto the sea in order to have a route of escape if the battle turned against the Normans.  Hardrada would not have to march his troops overland for 240 miles but could instead use his sthips to bring the rested troops almost to William's doorstep.  The Saxons and some of the Norsemen would be deployed to attack from the land side.  The Vikings, skilled at ship-to-ship fighting, would attack William's ships and attempt to gain entry to the castle from the seaward side.  Trapped in this pincers, William would be forced to surrender, pay a large ransom, and then be sent packing back to Normandy.
 

Hardrada's political and kinship ties with the budding kingdom of the Kyivan Russ would also result in contact between the Orthodox Church in the east and the Celtic Church in the north.  These contacts would be occuring less than two decades after the Great Schism that separated the Roman and Orthodox Churches; feelings would be running hot on both sides.  The Celtic and Orthodox Churches would stand united in their antipathy to interference and expansion from Rome.

The map at right shows that Irish (in blue) and Anglo-Saxon (in green) missionaries made a significant impact in Northern Europe.  These missions established great and influential abbeys and dioces whose Irish character was undiminished even as the majority of the monks came from the indigenous population.  With the new schism between the North and Rome, those missions (a) might have weakened Roman influence in the areas in centuries to come.

Until the coming of the Turks and the fall of the Byzantine Empire in the 15th century, the Roman Church would be boxed in geographically.
 

As mentioned above, Harold Hardrada had good connections with the Kyivan Russ (precursors to the Russian empire).  It stands to reason he and his heirs would attempt to solidify relations by political marriages spanning from England to the Ukraine.

The Crusades, if they still occured, would have a different impact upon the new England.  Instead of travelling through France, Germany, and Italy to reach Constantinople  and proceed to the Holy Land, English knights would sail through the North and Baltic seas and travel through Kyivan Russia.  (Follow the east-bound Viking routes on the map to see the connection.)

In spite of the presumed help provided to the Irish against the Franco-Normans, there is no reason to assume that the Anglo-Norse kingdom would not  pursue the conquest of Wales, Scotland, and Ireland (just as the Anglo-Norse did in reality).  But perhaps the Norse, not being quite as much an occupying alien as the Franco-Normans were, would feel less vulnerable and threatened by the presence of the Welsh and Scots on their borders and so would be less inclined to initiate more wars of conquest.  (Maybe, but I doubt it.)