The Standard Model of particle physics is our current best understanding of physics at the smallest scales. It assumes that, in the massless limit, the weak interactions of leptons are identical. This fundamental assumption is known as Lepton Flavour Universality (LFU) and can be tested by comparing the measured decay rates, or branching fractions ($B$), of (semi-)leptonic processes that differ only by lepton flavour. Such a test is summarised here, that compares the decay rates of $W$ bosons to either tau-leptons or muons, using $R(\tau/\mu)= B(W\to\tau\nu)/B(W\to\mu\nu)$. The measurement is performed with a novel technique using di-leptonic $t\bar{t}$ events based on 139 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}=13$ TeV recorded in the ATLAS detector at the LHC. Tau-leptons are identifed through their decays to muons. The lifetime of the tau-leptons provides two distinguishing features to determine whether muons originate from the $W$ boson decay or via intermediate tau-lepton: typically lower transverse momentum and a typically larger transverse impact parameter. The observed best-fit value of the ratio $R(\tau/\mu)$ is measured to be $0.992\pm0.013 [\pm0.007 \text{ (stat)} \pm 0.011\text{ (syst)} ]$ which is in good agreement with the Standard Model expectation of unity. This result achieves an unprecedented precision which is particularly promising coming from a hadron collider.