I'd say fairly important. Should you work in mechanical engineering after you graduate it will make your path to getting chartered considerably easier. However, a considerable proportion of graduates don't stay in engineering and even if you do it is possible to go through a process to satisfy the institution that you have the required knowledge and understanding even though your degree was not accredited. But...there are some big companies that only recruit engineers who have completed accredited courses (this is certainly true for Chem Eng, not certain for Mech Eng) so it may limit your employment choices later..
If it's a UK university it would be worth asking them why they're not accredited - almost all are.