Other nominees for the prestigious theatre awards include Dame Judi Dench, who is shortlisted for best actress, and Sherlock star Mark Gatiss whose performance in Coriolanus at the Donmar Warehouse sees him nominated for best supporting actor.
Dame Judi is nominated for her role in Peter And Alice.
Sam Mendes' musical of Charlie and the Chocolate Factory and the revival of Stephen Sondheim's Merrily We Roll Along have seven nominations each.
Law and Hiddleston, nominated for the title roles in Henry V and Coriolanus respectively, face competition from Henry Goodman and Rory Kinnear.
Radcliffe, who won best actor at the What's On Stage Awards for The Cripple of Inishmaan, is not shortlisted.