A time shift $\tau$ of the signal $\tilde x(t)$ can be implemented exploiting the time shift property of the discrete Fourier transform (DFT). So what you propose is possible.
Let $x(n) = \tilde x(nT)$ be the sampled version of $\tilde x$ with sampling interval $T$. The flanged signal $y$ can be obtained by
$$
y(n) = \operatorname*{IDFT}_N\left[\operatorname*{DFT}_N( x ) + \operatorname*{DFT}_N( x )\exp\left(-\mathrm j \frac{2\pi}{N}k\eta\right) \right]\\
= x(n)+ \operatorname*{IDFT}_N\left[\operatorname*{DFT}_N( x )\exp\left(-\mathrm j \frac{2\pi}{N}k\eta\right) \right],
$$
where $\eta = \tau/T$, $k\in[0,N-1]$, and $N$ is the length of the I/DFT.
The time shifted version of $x$ is just an interpolation if $\eta \notin \mathbb N$ and the precision of this interpolation grows with $N$. On the other hand, $\tau$ can only be changed every $N$ values and therefore the greater $N$, the lower the update speed. The value of $N$ is thus a tradeoff between time resolution and update speed.
For every block of $N$ samples an I/DFT pair is required which is computationally expensive.