I have a BEng and a CPL (never bothered with ATPLs as I've no great interest in going multicrew, if that changes, I'll go and do the extra exams), and have progressed fairly happily through life with the combination, and built on both.
In my opinion the combination of the two is absolutely great IF you have a deep desire to work on the boundaries of aviation and engineering. If you have an aspiration to be a test pilot, follow a military senior aviation career, or a researcher in aeronautical engineering (I've done two of those three, arguably all three if you'll accept running an ex-military research unit).
However, if you only want to be a civil pilot, don't waste your time and money doing an engineering degree. It is far more technical education, at far greater cost, than you will ever require.
If you really only want to be a professional engineer, then for now at least don't muck about with that level of flying training. Do a PPL, get on with your professional engineering training - and if you want more flying qualifications later, do those (and pay for those) whilst you're working as an Engineer.
If you want a backup, do something that makes you immediately employable and has a very long shelf life such as a swimming instructor or short order chef. 4 years of engineering, to give you the ability to *try* and enter a graduate training programme when you've already (from an engineering employer's viewpoint) wasted two years mucking about with aeroplanes while you forgot your engineering theory, is an incredibly high risk, and frankly stupid, strategy.