The red thread as a metaphor for a consistent theme is not unique to Swedish. It probably originated not with Theseus, but with Goethe who wrote in Elective Affinities:
There is, we are told, a curious contrivance in the service of the
English marine. The ropes in use in the royal navy, from the largest
to the smallest, are so twisted that a red thread runs through them
from end to end, which cannot be extracted without undoing the whole;
and by which the smallest pieces may be recognized as belonging to the
crown. Just so is there drawn through Ottilie's diary, a thread of
attachment and affection which connects it all together, and
characterizes the whole. 1
I couldn't find many instances of scarlet thread being used in the same sense in English. There is, however, a very famous speech made by Viscount Sankey in the landmark House of Lords case of Woolmington v DPP regarding the importance of the presumption of innocence in criminal law:
Throughout the web of the English Criminal Law one golden thread is
always to be seen that it is the duty of the prosecution to prove the
prisoner’s guilt subject to… the defence of insanity and subject also
to any statutory exception. If, at the end of and on the whole of the
case, there is a reasonable doubt, created by the evidence given by
either the prosecution or the prisoner… the prosecution has not made
out the case and the prisoner is entitled to an acquittal. No matter
what the charge or where the trial, the principle that the prosecution
must prove the guilt of the prisoner is part of the common law of
England and no attempt to whittle it down can be entertained. 2