Lets say William after taking England decides that the entire Saxon kingdom should go to him attacks Scotland wins and takes the Lowlands.
How will this affect the history of the British Isles?
Blackadder mk 2
If he did, although it was some time before he had full control of England, the biggest impact would probably be on Scotland. Since that would rob it of its richest area and the core of it's English speaking territory. As such, if it lost Lothian permanently it would be greatly weakened economically and militarily. Also probably revert to being a largely Gaelic speaking nation.
It might not be too simple however. The region was somewhat mixed in population and I expect that most of the English Lothians would have preferred Scottish to Norman rule. The former were sometimes savage and unpleasant neighbours but didn't have a major reason to persecute the population they ruled. While the Norman basic ethos was to dispossess and repress the local population.
One other point in that while Lothian was at the time the richest part of the lowlands they only covered the eastern part of it. The western part, around the Clyde valley was the core of the old British kingdom of Strathcylde, conquered by the Scots ~1018. This wouldn't be something the Normans could claim as part of England but then legality wasn't something that ever stopped them.
If Lothian was permanently regained by England and a English state re-emerged from the Norman period it would be a little stronger in absolute terms and relatively a lot stronger because Scotland would be that much weaker. This could mean more conflict as England tries to bring Scotland under control similar to Wales, or less as the smaller Scotland is seen as a threat and less capable of attacking northern England. The complicated factor might be that if Lothian is fully incorporated and hence the north is larger and stronger there might be greater regional tension inside England.
Steve