Oven spring is caused by the air pockets in the dough expanding from the heat. (Dough rises from gasses released from the yeast.)
After the shaping and final rise, often times there is a light, dry "skin" over the dough. By slashing a dough before it goes into the oven, you break this skin, and the bread is able to expand. If the loaf is a "fancy loaf" and you can't slash it without ruining the appearance (like a braided loaf), try to keep the loaf from drying out with a light mist of cooking spray after shaping and before the final rise.
Perhaps the bread is cooking too quickly when it hits the heat of the oven, essentially cooking a crust before the air pockets get heated enough to expand? Baking in a moist environment should help with that. Place an empty, sturdy pan (I use a cast iron skillet) on the bottom rack of the oven (or directly on the floor of the oven) before preheating. When you place the bread in the oven, pour about 1 cup of very hot water into the empty pan. This will create a bunch of steam, and help prevent the bread from crusting before it gets its "spring."
If you have a pizza/baking stone, use it. Having a hot surface to set your pans on helps with the rise. Think about it... you open the oven door, and out goes a lot of the heat... even though the walls of the oven are retaining the heat, it will take a little while before that heat reaches the bread. Setting the pan on the stone will give you that instant heat on the bottom, causing dough to rise from the bottom up, rather than just getting a small rise from the top area.
If you don't have a stone, invert a heavy baking sheet, cast iron griddle, or something similar, and heat that up in the oven the same as you would for a baking stone. Invert the sheet pan so it's easier to slide your bread pans on and off the hot pan without having to deal with a small edge.
If you haven't got equipment to do the other options listed above, you can try the "cold oven method." Just put the loaf into a cold oven, and set the temperature. Don't preheat. The gradual heat from the bottom of the oven as it preheats will give you some of that "oven spring."
I use a baking stone and steam, but I have had great success with using an inverted aluminum sheet pan with steam before I got the stone... and before I learned that trick, I used the simple cold oven method (no steam as the oven is cold!).
As long as you monitor each cake for doneness individually, there is no reason not to bake them together—assuming they both need the same temperature.
If the sizes are radically different (in terms of the thinnest dimension, which is usually the depth) you may need to adjust temperature to control crust formation during the time it takes the cakes through to the center.
If the cakes are close in depth, they should bake in approximately equal times.
You may wish to use cake strips or aluminum foil to protect the thing easily burned permiter of a hemispherical cake.
Note that 150 C (300 F) is an atypically low temperature for baking a cake.
See related: Additional Cooking Time When baking more than one loaf of quick bread
Best Answer
I'd guess it's not the new oven that's wrong, but the old one.
Older ovens has less accurate thermometers & were maybe 20° hotter at the top than the bottom. The chances are your hand-me-down recipe was based on this phenomenon & your new one is accurate… & therefore not hot enough.
I'd never heard of popovers until 10 minutes ago, but reading through recipes online, they seem to be exactly same as Yorkshire Puddings, just served with sweet toppings instead of roast beef & gravy. [How two sides of the Atlantic arrived at that difference, I'll never know nor understand ;)
Most say to use 450°F [230°C] which is more like I'd use for Yorkies - in fact for Yorkies, the hotter the better, preheat your tins too. My rule of thumb has always been, once everything else is done [roast dinner, remember;) just turn the heat to max, add your oil & put your tins in the top half [high as you can but leaving room to rise], then give it 15 mins to come up to temperature. The oil should be smoking before you drop your batter.
Pour quickly & get that door shut. Once in, never open the door until they're ready.
After comments
I've never known anyone to drop the temperature half-way through [but then again, 'crispy' is not something I'd want from a yorkie [only supermarkets & restaurants think they should be crispy, ordinary Yorkshire folks don't]), but I still think you're not getting enough heat into them right at the start.
Maybe your oven really does slump the temperature easily; maybe you've got the door open too long or the element isn't fast enough to get back up to temperature; tins aren't hot enough to start with.
I'd try at least once with the oven simply "on full" - whatever it thinks it can go to - but watch it doesn't switch mode right at the top. if I turn mine full then back it switches to rotisserie/fan grill rather than 'oven'. That might be a little excessive ;)