  The 21cm emission of neutral hydrogen is a potential probe of the matter
distribution in the Universe after reionisation. Cosmological surveys of this
line intensity will be conducted in the coming years by the SKAO and HIRAX
experiments, complementary to upcoming galaxy surveys. We present the first
forecasts of the cosmological constraints from the combination of the 21cm
power spectrum and bispectrum. Fisher forecasts are computed for the
constraining power of these surveys on cosmological parameters, the BAO
distance functions and the growth function. We also estimate the constraining
power on dynamical dark energy and modified gravity. Finally we investigate the
constraints on the 21cm clustering bias, up to second order. We consider the
effects on the 21cm correlators of the telescope beam, instrumental noise,
foreground avoidance, the Alcock-Paczynski effect and theoretical errors in the
modelling of the correlators. Adding Planck priors, and marginalising over
nuisance parameters, HIRAX achieves sub-percent precision on the $\Lambda$CDM
parameters, with SKAO delivering slightly lower precision. The modified gravity
parameter $\gamma$ is constrained at 1% (HIRAX) and 5% (SKAO). For the dark
energy parameters $w_0,w_a$, HIRAX delivers percent-level precision while SKAO
constraints are weaker. HIRAX achieves sub-percent precision on the BAO
distance functions $D_A,H$, while SKAO reaches 1-2% for $0.6\lesssim z\lesssim
1$. The growth rate $f$ is constrained at a few-percent level for the whole
redshift range of HIRAX and for $0.6\lesssim z\lesssim 1$ by SKAO. The
different performances arise mainly since HIRAX is a packed inteferometer that
is optimised for BAO measurements, while SKAO is not optimised for
interferometer cosmology and operates better in single-dish mode, where the
telescope beam limits access to the smaller scales that are covered by an
interferometer.
