Hey there, I study psychology at King's and got in with two A Levels (including psychology which was listed as one of the required subjects) and a BTEC Level 3 Extended Diploma. For my particular course this aligned with the entry requirements on the website. I also had an offer from UCL based on one A Level and my BTEC, but again it depends on the course and its specific requirements. I would just say don't be discouraged, it is absolutely possible. And I think doing a BTEC which is directly related to the degree can actually demonstrate more practical experience than A Levels which you could emphasise in your application. Best of luck!
Edit: keep in mind there may be different requirements based on how many credits your BTECs consist of, for example mine was a two year course and I got D*D*D*.