Can you clarify, what is the state of the unit during the 30-35 minutes before it begins functioning? Will it not turn on? Not responsive? Won't drive rudder indicator(s)? Won't drive pump? Etc?
It isn't uncommon for aging electronics to need a warm up time. And, it isn't uncommon for aging and failing electronics to need more of it. But, 30 minutes is excessive -- by a lot. Most things that aren't okay in a very few minutes don't magically become okay by not functioning for even longer. Even vacuum tubes didn't take 30 minutes to warm up. So, I'm actually going to bet that the problem isn't the unit, itself.
So, my first tests would be outside the box. Is the power good? All the way to the unit? Is the ground good? All the way up to the unit?
If the batteries have gotten tired or low on water, or the connections have gotten corroded and high resistance, or both, especially if wiring was too thin from the get go, it'd be really easy for the unit to get low voltage initially.
Then, as the boat warmed up, e.g. the alternators charged the batteries and voltage rises, other things pulled down from the initial start or a period of non-use get charged up and back to normal, the engines warm up the batteries, the charging warms up the batteries, etc, voltage might come up, allowing something that was threshhold before to start working.
As you can tell, my first guess is a temperature mitigated voltage issue. If all of the voltages look good, you can try opening the unit and heating it a little, gently with a hair dry on low setting and seeing if it helps. When I did electronics repair, we both heated and cooled components to try to isolate failures we couldn't reproduce the first time we looked.
But, I'm betting on voltage. Check carefully. Half checking doesn't often show results. If initially seem okay, check once initially and then again once things start working. See if the voltage at the unit (between its +12 and Neg) has changed and how they look.
Just my thinking.