Chances of getting into Oxford or Cambridge

I am entering my fourth year of undergrad and am considering applying to a masters program in the United Kingdom, including applying to Cambridge and Oxford. However, because of my lack of motivation in High School and early college, I am attending a less prestigious university and for my first year of college my grades were in the B average. However, going into my second year onward I have gotten straight As, with an A- thrown in every other semester. My GPA has recovered to the low 3.8 range where I am just under Magna Cum Laude. When I read the people applying to Oxford and Cambridge however, I see a lot more Summa Cum Laudes with high 3.8s and 3.9s, as well attending much more prestigious institutions, something I can never make up for because of past laziness. I know there is a possibility of being accepted to either Oxford or Cambridge, but how likely is that possibility when I look so weak academically compared to other applicants? I have several letters of recommendation and several smaller internships relating to my field, but that is all. Thank you all.

It entirely depends on which programme you are applying. Some are insanely competitive while others I’m sure you’d have a shot at.

agreeing with @quidgibo. It particularly matters how good a fit you are for the specific program: I know a recent applicant who had stunning credentials (GPA, awards, LoRs from leaders in her field), who was turned down at one but accepted to the other, b/c her research interests aligned well there.

Why are you focused on specifically getting into Oxford or Cambridge?