Note: this answer is no longer valid as it has been established that robots are not affectionate and, therefore, cannot hug or hold hands.
First Robot:
$$25 = v_{x1}*t_1$$ $$v_{y1}=a_g*t_1/2$$ $$10=v_{y1}*t_1/2-a_g*t_1^2/8=a_g*t_1^2/8$$ $$t_1=4\sqrt 5/a_g$$$$\begin{align} 25&=v_{x_1}t_1\\ v_{y_1}&=a_g\frac{t_1}2\\ 10&=v_{y_1}\frac{t_1}2-a_g\frac{t_1^2}8=a_g\frac{t_1^2}8\\ t_1&=\frac{4\sqrt 5}{a_g} \end{align}$$
Second Robot:
$$15 = v_{x2}*t_2$$ $$v_{y2}=a_g*t_2/2$$ $$20=v_{y2}*t_2/2-a_g*t_2^2/8=a_g*t_2^2/8$$ $$t_2=4\sqrt {10}/a_g$$$$\begin{align} 15&=v_{x_2}t_2\\ v_{y_2}&=a_g\frac{t_2}2\\ 20&=v_{y_2}\frac{t_2}2-a_g\frac{t_2^2}8=a_g\frac{t_2^2}8\\ t_2&=\frac{4\sqrt{10}}{a_g} \end{align}$$
Robots Hugging tightly and jumping at same time:
$$v_{xc}=\frac{m_1*v_{x1}+m_2*v_{x2}}{m_1+m_2}$$ $$v_{yc}=\frac{m_1*v_{y1}+m_2*v_{y2}}{m_1+m_2}$$$$ v_{x_c}=\frac{m_1v_{x_1}+m_2v_{x_2}}{m_1+m_2}\quad v_{y_c}=\frac{m_1v_{y1}+m_2v_{y2}}{m_1+m_2} $$
Assume the robots weigh the same (I can do more if you need):
$$0=v_{yc}*t_c-a_g*t_c^2/2=2(\sqrt {10}+\sqrt 5)t_c-a_g*t_c^2$$ $$t_c=2(\sqrt {10}+\sqrt 5)/a_g$$ $$v_{xc}t_c=(\frac{25}{8\sqrt 5/a_g}+\frac{15}{8\sqrt{10}/a_g})*2(\sqrt {10}+\sqrt 5)/a_g$$ $$v_{xc}t_c=21.49>20$$$$\begin{align} 0&=v_{yc}t_c-a_g\frac{t_c^2}2\\&=2(\sqrt{10}+\sqrt5)t_c-a_gt_c^2\\ t_c&=\frac{2(\sqrt {10}+\sqrt 5)}{a_g}\\ v_{x_c}t_c&=\left(\frac{25}{\left(\frac{8\sqrt 5}{a_g}\right)}+\frac{15}{\left(\frac{8\sqrt{10}}{a_g}\right)}\right)\frac{2(\sqrt{10}+\sqrt 5)}{a_g}\\ v_{xc}t_c&=21.49>20 \end{align}$$
