There are simpler solutions for special cases.
Definitions
$x_i=\mathcal{N}(p_i,\sigma^2)_{i=1...6}$ are 6 independent normal distributed random variates with expectations $p_i$ and variance $\sigma^2$ (this replaces $s^2$ from the original post).
The random variables are transformed by the Euclidian distance function $$L(x_0,x_1,x_2,x_3,x_4,x_5)=\sqrt{(x_0 - x_1)^2 + (x_2 - x_3)^2 + (x_4 - x_5)^2}$$ The distance of the expected values is $$L(p_0,p_1,p_2,p_3,p_4,p_5)=L_p=\sqrt{(p_0 - p_1)^2 + (p_2 - p_3)^2 + (p_4 - p_5)^2}$$
The expected value of $L$ is $\mathbb{E}[L]$.
Results
For following 2 cases simpler expressions of approximated expectations $\mathbb{E_1}[L], \mathbb{E_2}[L] $ are:
$$\frac{\sigma}{L_p} \approx 0 \hspace{2mm} \rightarrow \hspace{2mm} \mathbb{E_1}[L] =\sqrt{L_p^2+6\sigma^2} \approx \mathbb{E}[L] $$
$$\frac{L_p}{\sigma} \approx 0 \hspace{2mm} \rightarrow \hspace{2mm} \mathbb{E_2}[L]=\frac{4\sigma}{\sqrt{\pi}}\approx \mathbb{E}[L] $$
Derivations
The distribution $$L'(x_i)=\left(\frac{x_0 - x_1}{\sqrt{2}\sigma}\right)^2+\left(\frac{x_2 - x_3}{\sqrt{2}\sigma}\right)^2+\left(\frac{x_4 - x_5}{\sqrt{2}\sigma}\right)^2$$
is a noncentral chi square distribution with $k=3$ degrees of freedom and the noncentrality parameter is
$$\lambda=L'(p_i)=\left(\frac{p_0 - p_1}{\sqrt{2}\sigma}\right)^2+\left(\frac{p_2 - p_3}{\sqrt{2}\sigma}\right)^2+\left(\frac{p_4 - p_5}{\sqrt{2}\sigma}\right)^2=\left(\frac{L_p}{\sqrt{2}\sigma}\right)^2$$
The expectation value of a noncentral chi square distribution is
$$\mathbb{E}[L']=\lambda+k$$
If $\frac{\sigma}{L_p}\approx 0 $ then $L' \approx \lambda$ and using the approximation
$$\frac{1}{n}\sum_{i=1}^n \sqrt{\lambda+\delta_i}\approx \sqrt{\frac{1}{n}\sum_{i=1}^n (\lambda+\delta_i)} \hspace{3mm}\text{for}\hspace{3mm} |\delta_i|\ll \lambda $$
$\mathbb{E}[L]$ is approximated by
$$\frac{\mathbb{E_1}[L]}{\sqrt{2}\sigma}=\mathbb{E_1}\left[\sqrt{L'}\right]\approx \sqrt{\mathbb{E}[L']}$$
and it follows
$$\mathbb{E_1}[L] =\sqrt{L_p^2+6\sigma^2}$$
If $\frac{L_p}{\sigma}\approx 0 $ then $\lambda\approx 0$ and $\mathbb{E}[L]$ is approximated by the expectation of a chi distribution
$$\frac{\mathbb{E_2}[L]}{\sqrt{2}\sigma}\approx\sqrt{2}\frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)}=2\sqrt{\frac{2}{\pi}}$$
It follows
$$\mathbb{E_2}[L]=\frac{4\sigma}{\sqrt{\pi}}$$
It holds $\mathbb{E_1}[L] \ge \mathbb{E}[L] \ge \mathbb{E_2}[L] $.
Examples
If the values of $L_p$ and $\sigma$ are in the proper ratio then the approximations are quite close to the exact solutions. This is seen below where the approximate and exact solutions are compared. The example coordinates from another post $p_i=(1,0,2,0,3,0)$ are used, $L_p=\sqrt{14}\approx 3.74$.
- $\sigma=0.02, \frac{\sigma}{L_p}\approx 0$
$\small \mathbb{E_1}[L]=3.7419781$
$\small\mathbb{E}[L]=3.7418712$
- $\sigma=200, \frac{L_p}{\sigma}\approx 0$
$\small\mathbb{E_2}[L]=451.35167$
$\small\mathbb{E}[L]=451.36483$
$\small\mathbb{E_1}[L]=6.1644140$
$\small\mathbb{E_2}[L]=4.5135167$
$\small\mathbb{E}[L]=5.7275959$
Motivation
Why do we need these approximations?
The exact solutions require calculation of the mean of a noncentral chi distribution that practically means calculation of a generalized Laguerre function that is related to Kummer's confluent hypergeometric function. In Mathematica the functions $Hypergeometric1F1[a,b,z]$ with $a=-\frac{1}{2}, b=\frac{3}{2}$ or $LaguerreL[n,k,z]$ with $n=k=\frac{1}{2}$ can be used, $z$ has negative real part. There might be problems if you try to translate your code into another languange. Such special functions are not available in all software packages or supported only for $a,b,n,k\in \mathbb{N}$. Special treatment regarding numerical stability is needed if you write your own routine. So the given approximations may help in some cases.
p0, p1,...
ands
not known? $\endgroup$