In this context, finally would be used to indicate that finally, after a long and exhausting trip they will arrive. Lastly would indicate that they had been to a number of other places, and England is the last one they went to. However, the last place you go could also be described as a final destination. The difference is that 'finally' is an adverb, and 'final' is an adjective (although it is also used as a noun, as in the final of a sporting event). Hope that helps.

Finally sounds better. Eventually would probably be better still.