I think cycles can go straight on up the contraflow bus lane (it is a contraflow, right? if not go round the square)
It is a contraflow and you can ride on through.
My suggestion of the Bow Lane section is to avoid the nonsense that is Aldwych. It's not unridable but the switching lanes of traffic and likely pedestrian asshattery at the crossings doesn't fill me with joy. Riding up past the Opera House and then around by the Museum means you get a chance to check if anything good is going on.
It is a contraflow and you can ride on through.
My suggestion of the Bow Lane section is to avoid the nonsense that is Aldwych. It's not unridable but the switching lanes of traffic and likely pedestrian asshattery at the crossings doesn't fill me with joy. Riding up past the Opera House and then around by the Museum means you get a chance to check if anything good is going on.