Some references

• "Berry-Esseen bounds and almost sure CLT for the quadratic variation of a general Gaussian process"
• "Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion"
• "Berry–Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion"

They study this rate of convergence for fractional-BM. For the particular case of standard Brownian motion, we have

$$\frac{V^{n}_{t}-t}{\sigma\sqrt{n}}\sim N(0,1),$$

where $V^{n}_{t}=\sum^{n}_{k=0} (B_{k+1}-B_{k})^{2}$.

In "Bounded Laws of the Iterated Logarithm for Quadratic Forms in Gaussian Random Variables", they at least prove the upper bound in Theorem 3.1.

$$\limsup_{n}|V^{n}_{1}-1|/\phi_{n}\leq 1,$$

for $\phi_{n}:=\sqrt{2}E[(V^{n}_{1})^{2}]\log\log(1/E[(V^{n}_{1})^{2})$. And as mentioned in remark 3, we also have

$$\limsup_{n}|V^{n}_{1}-1|/\phi_{n}>0,$$

for some special partitions (fast decay $t_{i}=\frac{i}{2^{3n}}$).

Since you are interested in various divisions, here are some more references

• "On almost sure convergence of the quadratic variation of Brownian motion"
• "Sufficient and Necessary Conditions for Limit Theorems for Quadratic Variations of Gaussian Sequences"
• "Quadratic variation for Gaussian processes and application to time deformation"