Generally, we can expect the English economy to expand in the same way under the continuing native English kings as it did under the Norman and French kings. This will allow a similar expansion of power into other corners of the British Isles, for which there is a precedent in pre-1066 England. In the 10th century, the house of Wessex was very involved in Scotland (or, what would become Scotland), and there were several invasions under different kings. King Aethelstan in particular comes to mind as an English king who wielded great influence throughout the British Isles.
The first challenge the English are likely to face is the subduing of regional tendencies in the former Northumbria, which can perhaps be achieved by the middle of the 12th century? William the Bastard achieved it much quicker with brutal methods, but I can't imagine the native English resorting to genocide and terror to solve this problem on the same scale he did.
It seems pretty common to assume that England in this situation will tend much more towards Scandinavia than OTL. The reasons given are just ones of cultural similarity and some fading political ties. Really, the proximity of England to the continent is going to make it much more likely that England continue to integrate into the Europe as it did after the Norman conquest and was doing before it. Remember that there were already heavy Norman influences in the English court in the reign of Edward the Confessor, and England was by no means isolated from the continent. Add to that the increasing volume of trade we can expect as the Middle Ages progress, which will link England economically to Flanders and other continental regions regardless of political developments.