From what I can see there are non-numeric parameters. How is NSolve expected to deal with them?

It would also be useful to have some description of the geometry problem under consideration. The fourth equation is to impose colinearlty with point 2 in the middle? The last two equations are some sort of equal slope constraints? Maybe it would make sense to change coordinates so that all points lie on the x axis? If so, that would simplify the system substantially.

If you give up on the current calculation, and I am not suggesting that you do that, then

You have three equations with square roots on both sides. You can square both sides and eliminate all, but one of those square roots. If you find solutions to this revised set of equations, you must then check to see which solutions satisfy the original equations.

You have two equations with denominators on both sides. You can cross multiply and eliminate those denominators. If you find solutions to this revised set of equations, you must check to see which solutions satisfy the original equations.

You are asking NSolve to only find Real solutions. I believe I have seen at times the calculation can take substantially longer when limiting solutions to only be Reals. If you remove the Reals limit and find solutions then you must check to see which solutions satisfy your condition.

I do not know how much time, if any, this might save, but those would be the things that I would try. I do not see other things in your notebook that make me think they are a problem.