The technology you're describing is about WWII era. With light bulbs, radios, and motors you can actually do quite a lot. Specifically the inclusion of radios mean that you have vacuum tubes or other such amplifier units, meaning that you actually have most of the simple building blocks of navigational equipment, as well the manufacturing process (such as aluminum smelting) that provide some of the semi-advanced air frames that make spaceflight easier.
The main limitation on your spacecraft, as well as the main feature in construction, will be the lack of computer aided design that has shaped the modern aerospace industry. Dust off your slide-rule and get ready for some 3rd and 4th order differential equations!
Now, hand calculation doesn't sound all that bad (unless you don't enjoy math), but it will have two noticeable effects.
1) Your space vehicle will not be optimized! Hulls will be unnecessarily thick, fuel systems will be less efficient, components will require larger tolerances, etc. If you want some case studies I recommend reading up on WWII era fighters, bombers, and freight aircraft. At the very least use google images and find some visual inspiration from technology of that era.
2) Your space vehicle will require larger crews. Without low-level automation of maintenance tasks, you'll need skilled laborers to keep the ship running smoothly. Additionally, you'll need specially trained navigators and pilot crews to fly the vehicle. A secondary effect on crew size is that your ship--which is not optimized and like an early Soviet ICBM will have to be very over-sized to achieve it's goal--will have to be larger, carry more fuel, require more maintenance, and so forth. This larger size will require a larger crew to maintain and control.
As a side note on computational systems, though, you don't need heavier and harder to maintain digital systems like fluidics or pneumatics to build a computer. If you have radios, you have an amplifier element--the buidling block of a digital switch. Sure, your computer will be the size of a large apartment, but if its fast navigational calculations your trying to run, it may be worth lugging UNIVAC around.
I should note too, as relates to your comment on using solid fuel, that a liquid fueled rocket is very much within the reach of a non-electric society. Also, if you've got motors, it is fairly easy to due fly-by-wire systems with minimal knowledge of how electricity work. The components used in a simple feedback loop like PID controller were available before WWI and where actually tested in a auto-pilot like drive-by-wire system for seagoing vessels. As for the air-supply, all you need is a fan to circulate air, a couple of plates of compressed and chemically processed volcanic ash and some oxygen producing reaction and you've got a low-tech scrubbing system that worked for pre-WWI U-boats BUT is still in use on modern spacecraft.
That fact that your civilization is making it to the exosphere before the advent of digital computers is impressive, but it makes me think that your probably avoiding the development of the digital age, and that the current state of aerospace engineering is the result of a long process of trial and error. That said, the technology you've made available makes it possible to build a surprisingly sophisticated space-craft, so enjoy going inter-planetary (or least back and forth from the space-resort 150 miles above the Adriatic Sea)!
Hope that Helps!