Hey there,
As far as your scores are concerned, the GPA may be a bit less for LSE and Oxbridge. A competitive GMAT score can compensate for this, so you may want to check out the median GMAT scores of these universities.
Aim for a 700+ GMAT score to get a fair shot at your target schools. Also, it would be great if you have a relevant reason for your low CGPA You can use your essays to justify the same.
Overall, your profile looks fine – You have decent amount of internships in diverse industries. You have not given much details on your extra-curriculars, but as you have mentioned that they are fairly good, we can take it on the face value.
You need to have a good story to supplement your profile, hence the importance of essays is paramount in your application. Use your essays to showcase what strengths, values and competencies you bring to the table.
As far as your target schools are concerned HEC, LBS,and Oxbridge are very selective programs; they belong to your dream category.
With good essays and a 680+ GMAT, you should have a fair shot at these LSE and Bocconi.
Also, note that you should target schools on the basis of where you want to work after your degree. If it is US,UK, France or any other country for that matter, analyse the job prospects and then shortlist your target schools.
Check out our FREE PROFLE EVALUTION to get a realistic idea on your profile and selection chances:
https://mim-essay.com/profile-evaluation?gcf