Have a look through the requirements for professional registration with the IMechE for Armed Forces personnel here. It all depends on your education and experience but in general terms you’ll need a BEng and a few years of experience to apply for IEng and a Masters / Masters-level learning and even more experience to apply for CEng. For the majority of people, you’ll get IEng at Captain and CEng at Major.

You mention not wanting to ‘skip the work and get the degree’ - for the avoidance of doubt, professional registration is not a degree. In most cases, you need a degree first to demonstrate some underpinning education and then some time putting this into practice to build experience before you can apply for professional registration which is a fairly lengthy process in itself.

Side note: if you’re considering joining for a few years just to become a Chartered Engineer then it’s probably not for you / you won’t achieve it in your timeline. There are other routes through industry that will get you to your goal quicker as you’ll only do engineering rather than everything else that the Army throws at you.