There is no hope of making a justifiably good estimation!
Lacking any details about the path inbetween, an adequate guess would be to approximate the curve with a cubic Bezier curve, where $P_0$ is the starting point, $P_3$ the end point, and $\vec{P_0P_1}$, $\vec{P_2P_3}$ are proportional to the measured velocities. This still leaves one degree of freedom, so maybe we should pick the scale so that the length $|P_1P_2|$ is "compatible" with $|P_0P_1|$ and $|P_2P_3|$, for example their geometric mean. In principle, this finally determines a curve the length of which can be computed.
However, the details of the computation are possibly more complicated than the little available input can justify.
So alternatively, especially in a situation like your example where the curve must have a point of inflection, one could try to fit in two touching circles with radii proportional to the measured speeds. However, this does not produce a unique curve either (though the length computation should be simpler)
So finally a very crude (or: quick and dirty) method: When moving along a line at an angle of $\delta$ against the straight connection, the effective "progress" in the intended direction is shorter by a factor of ${\cos\delta}$. Now let's simply assume that about $\frac13$ of the time, we move parallel to the direct connection (at a speed between the two measured ones), $\frac13$ of the time we move at the first measured speed and at a deviation against the straight line as measured there, and the remaining $\frac13$ of the time we move at the second measured speed and at a deviation as measured there.
In your example, the "progress" made per $t$ seconds would be
$$\tag1\frac t3\cdot(25\cdot\cos(220^\circ -245^\circ) +\tfrac{25+10}{2}+10\cdot \cos(230^\circ-245^\circ))\approx 16.605\cdot t$$
and the actual distance traveled along the curve in the same time would be
$$\tag2\frac t3\cdot(25+\tfrac{25+10}2+10) =17.5\cdot t$$
We need to pick $t$ so that $(1)$ produces the given distance of $15$ meters and then $(2)$ shows the result. This way we estimate the curve length as
$$ \frac{17.5}{16.605}\cdot 15\,\text{m}\approx 15.8\,\text m.$$
As said above, this method is very quick (really not much calculation involved) and dirty (approximations made that could make a physicist happy, but certainly not a mathematician) and I won't rely on it in cases where
the start or end dirction deviate substantially from the straight direction (the $25^\circ$ as in your example may still seem reasonable, maybe even up to $45^\circ$) and/or where the initial and final speed differ substantially (your example has a factor of $2.5$, which may or may not be considered reasonable; certainly a factor of $10$ would make the estimate totally untrustworthy).
