Typical of the gyratory systems of the 1960s that are now largely discredited and a hostile environment for cyclists: at some point you will probably have to cross at least two lanes of fast-moving traffic. Not to mention the noise, fumes etc.
As part of the London Bus Priority network, there was a scheme to introduce a contraflow westbound bus lane in the Aldwych and to reduce the width of the parallel section of the Strand. Some of the released space would become a pedestrian area and a service road.
When the Transportation and Highways Committee discussed the scheme on 23/01/01, members delayed further progress on the scheme (and destroyed any benefit for sustainable forms of transport) by asking officers to ensure that no group of road users would be given an advantage over other groups of road users.
In the summer of 2014, the Northbank Business Improvement District carried out a study of their area and sought to start discussions with Westminster and TfL officers on the options for reviewing the current gyratory at Aldwych.
In the summer of 2015, the orthbank Business Improvement District unveiled proposals to replace the Aldwych gyratory with two-way roads and create new spaces for pedestrians and cyclists.
In the meantime, your best way to avoid the Aldwych eastbound is to use Long Acre, Great Queen Street and Lincoln's Inn Fields. Westbound is more difficult, though the Victoria Embankment is an alternative.