What's the duration between the binding of adrenaline to β-AR and the first translated protein?
Things can certainly vary for different ligands and different levels of cascading but these are the reactions that you may consider.


Receptor activation
Nuclear translocation (lets assume it is just a part of the receptor. You can assume secondary messengers; it will add an extra step)
DNA-binding and Transcription activation
Transcription
Nuclear export of mRNA
Translation


Finally I guess what you really want is to know when steady state is reached. This will also depend on other factors such as:


Receptor deactivation rate
mRNA and protein degradation rates
Nuclear export of activated TF
Network architecture (some networks can speed up response times). This factor can be ignored and we can consider an open loop.


Phosphorylation reactions are generally much faster than transcription/translation/translocation rates and can also assumed to be equilibrium with dephosphorylation reactions.

Nuclear import rates are also dependent on cell volumes and for polarized cells like neurons it will be much higher. The average import rate is around 0.2 - 1.2 s-1 NPC-1 µM-1(cargo) [where NPC is number of Nuclear pore complexes which is again on an average 120 ] [1]. 
It can be, for ease, assumed to be ~0.02 min-1 [2].

In most modeling papers the transcription rate is assumed (based on experimental reports) to be 1 per min (Average polymerase speed of ~1000nt/min and gene length of 1000nt. This can vary with gene length)

Translation rates are somewhat trickier. The translation rate per ribosome translation elongation rate is ~5 codons/sec (15aa/sec)[3] or perhaps may even be lower — 3 - 9aa/s[4] ( [3] is a more recent finding). However, the overall translation rate will be dependent on number of ribosomes (Ribosomes can be assumed to be infinite unless it is some condition like stress or starvation. By no of ribosomes I meant polysomes per mRNA, which can also vary). Also, the protein synthesis rate will depend on length of mRNA. This is one parameter that can have a broad range of values (0.6-100 min-1 × mRNA). 

You can set up mathematical model (based on reaction rate equations) and calculate the response times (Depending on parameters it can be 8-20 hours). Steady states can generally reach by 8 hours (based on my own experiment. But I used tetracycline to induce luciferase). Another experiment (unpublished) using IFNγ induction caused a peak in the reporter at ~12h. I am not linking all references (you can easily look up for these). Usually, for cell culture applications, there is some sort of unsaid rule to assume 12h as a general response time to persistent stimuli. Response times are also dependent on initial conditions. 

Note: Response time means the time required to achieve steady state. This does not mean the delay between signal onset and protein synthesis. Other metric that is used generally is max-peaking time.