eqns = {a + b + c + d == 4*m0, b + d == 4*m1, c + d == 4*m2, d == 4*m3} /. {a -> t0/(1 + t0), b -> (t0*t1)/(1 + t0*t1), c -> (t0*t2)/(1 + t0*t2), d -> (t0*t1*t2*t3)/(1 + t0*t1*t2*t3)}

Solve[eqns, {t0, t1, t2, t3}]

sol = {t0 -> a/(1/4 - a), t1 -> (b/(1/4 - b))*((1/4 - a)/a), t2 -> (c/(1/4 - c))*((1/4 - a)/a), t3 -> (m3/(1/4 - m3))*(a/(1/4 - a))*((1/4 - b)/b)*((1/4 - c)/c)} /. {a -> m0 - m1 - m2 + m3, b -> m1 - m3, c -> m2 - m3}

eqns /. sol // Simplify

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Solve[{a + b + c + d == 4*m0, b + d == 4*m1, c + d == 4*m2, d == 4*m3} /. {a -> t0/(1 + t0), b -> (t0*t1)/(1 + t0*t1), c -> (t0*t2)/(1 + t0*t2), d -> (t0*t1*t2*t3)/(1 + t0*t1*t2*t3)}, {t0, t1, t2, t3}]