I use Birmingham airport frequently. My son tells me that it has a dedicated seating area for me. I sometimes go straight to work from the airport after weekend in Europe. Here is my advice.
If you are using domestic flights, that includes flights from Dublin, you do not need to go through the passport control. That save you time.
But if you have a baggage to claim, it can take time. Remember that even if you have a hand luggage, airline sometimes take it and put it to the checked in baggage area. The way to avoid is to be at the front of the line and use all your charm if they tried to take it out from you.
You take the train from the airport to the railway station. If you know the platform, then you can leg it quickly down the steps.
It takes maximum 10 minutes from leaving the airport to the platform. Yes I have timed it.
If the flight arrives in time, you can do it. But I know the airport and station as the back of my hand and do not wait to look for signs etc.