Your equation is easier to handle if it put into an equivalent form.

eq1 = Sqrt[l1] + Sqrt[t1] + Sqrt[w1] == d;

It is also to convenient to write the rules as

r1 = {l1 -> (pw^2 w1)/((1 + a e1)^2 w^2)};
r2 = {t1 -> (pw^2 w1)/((1 + a e1)^2 r^2)};

and substitute u^2 for w1 getting a new equation

eq2 = eq1 /. Join[r1, r2] /. w1 -> u^2

Sqrt[u^2] + Sqrt[(pw^2 u^2)/((1 + a e1)^2 r^2)] + 

  Sqrt[(pw^2 u^2)/((1 + a e1)^2 w^2)] == d

This can be simplified

 eq3 = Simplify[xpr1, Assumptions -> {u > 0}]

d == u (1 + Sqrt[pw^2/(r + a e1 r)^2] + Sqrt[pw^2/(w + a e1 w)^2])

Solving is now trivial.

soln = Solve[eq3, u][[1]] // Simplify

{u -> d/(1 + Sqrt[pw^2/(r + a e1 r)^2] + Sqrt[pw^2/(w + a e1 w)^2])}

Finally, to get w1 it is only necessary to evaluate u^2.

w1 = u^2 /. soln; w1

d^2/(1 + Sqrt[pw^2/(r + a e1 r)^2] + Sqrt[pw^2/(w + a e1 w)^2])^2

Your equation is easier to handle if it put into an equivalent form.

eq1 = Sqrt[l1] + Sqrt[t1] + Sqrt[w1] == d;

It is also to convenient to write the rules as

r1 = {l1 -> (pw^2 w1)/((1 + a e1)^2 w^2)};
r2 = {t1 -> (pw^2 w1)/((1 + a e1)^2 r^2)};

and substitute u^2 for w1 getting a new equation

eq2 = eq1 /. Join[r1, r2] /. w1 -> u^2

Sqrt[u^2] + Sqrt[(pw^2 u^2)/((1 + a e1)^2 r^2)] + 

  Sqrt[(pw^2 u^2)/((1 + a e1)^2 w^2)] == d

This can be simplified

 eq3 = Simplify[eq2, Assumptions -> {u > 0}]

d == u (1 + Sqrt[pw^2/(r + a e1 r)^2] + Sqrt[pw^2/(w + a e1 w)^2])

Solving is now trivial.

soln = Solve[eq3, u][[1]] // Simplify

{u -> d/(1 + Sqrt[pw^2/(r + a e1 r)^2] + Sqrt[pw^2/(w + a e1 w)^2])}

Finally, to get w1 it is only necessary to evaluate u^2.

w1 = u^2 /. soln; w1

d^2/(1 + Sqrt[pw^2/(r + a e1 r)^2] + Sqrt[pw^2/(w + a e1 w)^2])^2