  If cosmic strings are formed in the early universe, their associated loops
emit gravitational waves during the whole cosmic history and contribute to the
stochastic gravitational wave background at all frequencies. We provide a new
estimate of the stochastic gravitational wave spectrum by considering a
realistic cosmological loop distribution, in scaling, as it can be inferred
from Nambu-Goto numerical simulations. Our result takes into account various
effects neglected so far. We include both gravitational wave emission and
backreaction effects on the loop distribution and show that they produce two
distinct features in the spectrum. Concerning the string microstructure, in
addition to the presence of cusps and kinks, we show that gravitational wave
bursts created by the collision of kinks could dominate the signal for wiggly
strings, a situation which may be favoured in the light of recent numerical
simulations. In view of these new results, we propose four prototypical
scenarios, within the margin of the remaining theoretical uncertainties, for
which we derive the corresponding signal and estimate the constraints on the
string tension put by both the LIGO and European Pulsar Timing Array (EPTA)
observations. The less constrained of these scenarios is shown to have a string
tension GU < 7.2 x 10^{-11}, at 95% of confidence. Smooth loops carrying two
cusps per oscillation verify the two-sigma bound GU < 1.0 x 10^{-11} while the
most constrained of all scenarios describes very kinky loops and satisfies GU <
6.7 x 10^{-14} at 95% of confidence.
