It's easier to work with the equivalent formulae
$$ \sin{(u+v)}+\sin{(u-v)} = 2\sin{u}\cos{v} $$
and so on. We can proceed in a similar way to the trigonometric formulae, using the addition formula
$$ \wp(u \pm v) = \frac{1}{4} \left( \frac{\wp'(u) \mp \wp'(v)}{\wp(u)-\wp(v)} \right)^2 - \wp(u) - \wp(v) $$
(the minus sign follows from the plus sign formula since $\wp$ is even and $\wp'$ odd). Then
$$ \wp(u+v)+\wp(u-v) = \frac{\wp'(u)^2+\wp'(v)^2}{2(\wp(u)-\wp(v))^2} -2\wp(u)-2\wp(v), $$
which can also be written as
$$ \wp(u+v)+\wp(u-v) = \frac{(\wp(u)+\wp(v))(2\wp(u)\wp(v)-g_2/2)-g_3}{2(\wp(u)-\wp(v))^2} $$
using the differential equation, whereas
$$ \wp(u+v)-\wp(u-v) = -\frac{\wp'(u)\wp'(v)}{(\wp(u)-\wp(v))^2}. $$
(In fact, it's easier to prove these two than the addition formula: see e.g. here for my derivation of the addition formula using these.)

Another example is the formula
$$ \wp(v)-\wp(u) = \frac{\sigma(u+v)\sigma(u-v)}{\sigma(u)^2\sigma(v)^2} $$
(https://dlmf.nist.gov/23.10.E3), which uses the quasi-periodic $\sigma$-functions as well.

Any formula for $\wp'$ can be found by differentiating one for $\wp$, so they are not normally considered separately.

But $\wp$ isn't really much like $\sin$: it's more like $\pi^2/(4\omega_1)^2 ( \csc^2{(\pi z/(2\omega_1))} -1/3 )$, which we probably don't really expect to have a nice addition formula.

$\DeclareMathOperator{\sn}{sn}
\DeclareMathOperator{\cn}{cn}
\DeclareMathOperator{\dn}{dn}
$

A closer equivalent is the Jacobi elliptic function $\sn$, which has the addition formula
$$ \sn(u \pm v) = \frac{ \sn{u}\cn{v}\dn{v} \pm \sn{v}\cn{u}\dn{u} }{1-k^2\sn^2{u}\sn^2{v}}, $$
and hence
$$ \sn{(u+v)} + \sn{(u-v)} = \frac{ 2\sn{u}\cn{v}\dn{v} }{1-k^2\sn^2{u}\sn^2{v}} \\
\sn{(u+v)}-\sn{(u-v)} = \frac{ 2 \sn{v}\cn{u}\dn{u} }{1-k^2\sn^2{u}\sn^2{v}} $$
Since $\sn \to \sin$, $\cn \to \cos$, $\dn \to 1$ as $k \to 0$, these obviously become the trigonometric addition formulae. Similar ones exist for the other Jacobi functions.