The standard 240V water heater only has one speed (choice of amp draw): 23A. That is how all such hot water heaters are sized. The reason is they want it as fast as possible while falling within a statutory 24A limit.
Edit: ThreePhaseEel brings up an interesting point. A 30A hot water heater implies that it is 240V, because it's cheaper to bump volts to 240 before amps to 30. However, you didn't tell us the old service was 240V (meaning a 2-pole "double" breaker in the panel). If this is in fact a 120V heater, that changes things. I'll describe this difference inside this box.
First, make sure this is a "plain old resistive heater" and not a hybrid or heat-pump heater. The trick I'm about to describe won't work with a hybrid.
Now I said "resistive" - the heating element is a resistor, which is the simplest electronic device on earth, and it follows simple rules, starting with Ohm's Law: Voltage = Current x Resistance. Resistance is defined by how the heater is built. So what happens if we run it on 120V?
Well, since Resistance is fixed -- if Voltage drops by 50%, so must Current. That means on 120V, this same water heater will draw 11.5 amps, and will be within safety limits of a 15A/120V circuit, and will not trip breakers. However since useful power = voltage x current, and both those are halved, useful power is 1/4.
What does that mean for heater performance? Assuming it's Energy Star, the insulation is tip top. It will reach target temperature ........... eventually. And it will take 4 times longer to get there. You'll want to follow the conservation tips in my last paragraph.
This use on 120V is safe, and it's safe indefinitely.
Now back to the 240V scenario.
All of the advice to follow violates Code. The only Code-correct solution is correct cable and breaker, but you knew that.
So expect a lot of breaker trips.
Wait 1 hour before resetting it. That is because of this particular situation - we know exactly why it is tripping, so the consequences of resetting it too soon are also known.
If you upgrade the cable to #10 copper (10/2 will suffice), then you can reset it as often as you please. I am assuming perhaps it is hard to obtain the right breaker due to obsolescence (PushMatic) or complexity (quadplex).
If you have a Federal Pacific, Zinsco or Challenger panel, those panels are dangerous and you are already playing with fire, so don't push your luck with those panels.
You will have fewer trips if you take great pains to use hot water VERY sparingly - follow California drought or boondocking-in-RV rules. Get wet, shut the water off, lather up, then do a quick rinse... Select cold water in dishwasher and washing machine... Let the soap do the cleaning not the heat... Etc.