Instantaneous DC output using Laplace transform
What you have are: two 1st order lag terms, and a 2nd order underdamped term with the slight complication of a 2nd order numerator.

The general ROT is that the settling time for a 1st order lag is \$4\tau\$ where \$\tau\$ is the time constant (\$\tau\$ is the coefficient of \$s\$ in the general term: \$1/(1+\tau s)\$). The settling time for a 2nd order denominator term is given by: \$4/(\zeta \times \omega_n)\$ where \$\zeta\$ is the damping coefficient and \$\omega_n\$ is the natural frequency.

One standard form for the 2nd order denominator is: \$1 + 2(\zeta/\omega_n)s + s^2/\omega_n^2\$

In your case \$\omega_n=150\$ rad/sec and \$\zeta=0.4\$, hence the settling time for this term is \$4/(0.4 \times 150) = 0.067 = 67\$ms. Note, for this particular application we can ignore the 2nd order numerator as it only has a small effect on the transient generated by the denominator.

The two 1st order terms give settling times: \$4 \times 1.2 = 4.8\$ms, and \$4 \times 8 = 32\$ms. The longest settling time is thus \$67\$ms, after which all the transients may be considered spent.

The steady-state gain (or 'DC gain') of the sensor is unity. This can determined by setting \$s = 0\$ in the TF.

Here's a useful plotting tool for time domain and frequency domain responses from Laplace TFs:

http://sim.okawa-denshi.jp/en/dtool.php