Nope, you do NOT have to have a BEng or MEng to become chartered. The most important thing is the accreditation of the degree. If a professional institution accredits the BSc, then, upon further work experience and more academic study (though not necessarily at university), you can rise to chartered standard.
You generally need to have a masters level equivalent (note an MEng is not considered an MSc equivalent, but thats another argument for another day) and at least 2 years work experience in areas which your institution requires for you to submit your chartership application. So with a BSc, you'd need to get further accredited academic qualification, be that an accredited MSc or other diploma, etc, and also relevant work experience which ticks all the boxes for your institution. On the Eng side of things, an MEng, provided it is accredited, is usually sufficient to tick the academic part of chartership off. A BEng is not. But you'd still need industrial experience on top, usually 2 years with a Masters.
I think your best bet is to contact the institution directly- examples of istitutions are IET, IMechE, IMarEST, RAeS or the IChemE, but that is by no means an exhaustive list.
In Summary- chartership requires two sections: 1)ACCREDITED academic study. usually MSc or MEng equivalent 2)industrial work experience. typically 2 years, but can be longer.
Stu Haynes MEng MIET