Expanding my comment on the threeway intersections, we get an auxiliary result:

**Lemma 1.** If $v>5$, then starting from a threeway intersection where the branches have $0$, $a$ and $b$ units of weed, and enough free space outside that, the mower can clear the intersection in time $2(a+b) \, / \, (v-5)$.

(By "enough space" we mean that any other weed segments will not touch these branches during the operation.)

**Proof.** Keep doing round-trips to the three branches and back alternatively, always choosing the longest branch. Let us first consider the first round-trip and assume without loss of generality that it is $b$.

 - The outgoing leg takes time $t=b/(v-1)$ because the mower starts $b$
   units behind the tip, and his relative speed is $v-1$ as the tip is
   growing.  During this time, the other two branches grow $t$ units
   each.
 - The incoming leg also takes time $t$.  During this time
   the mower completely cleans this branch (including any weed that was
   growing back to the branch from the intersection). The other branches
   grow another $t$ units each.

Overall in one round-trip, in $2t$ time the net weed decrease is $b - 4t = t(v-5)$, so weed is cleared at rate $(v-5)/2$.  It is easy to see that the same average rate holds for each round trip.  Having started with $(a+b)$ units of weed, it is all cleared in time $2(a+b) \, / \, (v-5)$.

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If $v$ is large — let's take $v=100$ for concreteness — this suggests the following strategy. Let $L,R$ be the left and right intersections.

- Start from $L$.  Make three round-trips to $R$: first along the north route ($3$ units and back), then along the middle route ($1$ unit and back), and then along the south route ($3$ units and back). In total this takes time $0.14$.
- Now $L$ has weed lengths north=$0.08$, middle=$0.06$ and south=$0$.
- And $R$ has weed lengths north=$0.11$, middle=$0.07$ and south=$0.03$.
- By Lemma 1, we can clear around $L$ in time $2\cdot 0.14 / 95 < 0.003$.  During this time the weed around $R$ grows less than $0.01$ in each direction.
- Move east to $R$ in time $0.01$.  During this time the weed grows another $0.01$ units.
- Now $R$ has weed lengths north $< 0.13$ and south $< 0.05$.
- By Lemma 1, we can clear around $R$ in less than $2 \cdot 0.18 / 95 < 0.004$ time.

Clearly $v=100$ suffices.  It should now be relatively straightforward to find the minimum $v$ that suffices *with this strategy*.  Of course this says nothing about possible other strategies.