Likelihood of getting an offer at an undergrad Psychology course?

For Oxford - I would never say never but not having a science, given it is highly recommended, would probably put you at something of a disadvantage. Psychology at Oxford is very much taught as a science and they want applicants to appreciate it as a scientific discipline so you'd have to very strongly demonstrate that in the rest of your application. I know quite a few people who got in with 2 humanities a levels + Psychology but don't think I've met anyone at Oxford doing EP who didn't study any science a-level. Looking at Cambridge it seems like a lot (but not all) colleges want you to have A level Biology or Maths so you may also be at a disadvantage there. Do you also have at least a 7 in maths GCSE - that's important for Oxford too. If you are dead set on Psychology is there any chance your college would let you swap one of your A levels to a science?

I don't think it is impossible for you to study Psychology at Oxbridge with no science A levels, but it will be much much harder for you to get in.

If you can't and end up not getting onto a Psychology course, there are other routes into the field. E.g. HSPS at Cambridge allows you to study some Psychology modules (though the degree would not be BPS accredited), and you can do a Psychology conversion masters after your undergrad to get BPS graduate basis for membership if you're interested in ultimately going into practicing psychology (eg as a clinical psychologist).