If you're specifically not choosing to become a psychiatrist, then anything else is fine. Psychiatry will require a degree in medicine, in which case you would ideally have biology and chemistry in your arsenal.
If you want to become a therapist, then you would be looking to do a BPS accredited psychology related degree (not that you would need a degree for some areas of psychology). A number of these degrees don't have specific subject requirements i.e. you can get 3 A Levels in any subjects and still get in. The pickier degree courses would ask for psychology or biology as a required subject. In other words, biology or psychology alone would be more than sufficient as your options (the other 2 A Levels can be in anything you want, so long you get high grades in them).
To look for BPS accredited degrees, see:
https://portal.bps.org.uk/Accredited-CoursesIf you specifically want to be a counsellor, you would ideally be doing a Level 3 course in counselling to get in. Whilst the industry doesn't strictly require you to have anything to call yourself a counsellor, a Level 3 is recommended.
Should you want to do something at uni to go into counselling, you can still go into the field with a degree in psychology, as opposed to just doing a degree in counselling. Personally, I would recommend the BPS accredited psychology degree, because you can more or less do anything in psychology with it (except psychiatry).
Should you decide to do a degree in counselling for any reason, the general entry requirements are 3 A Levels in any subjects, so long you meet the minimum grade requirements.
Note:
Chemistry A Level is one of those subjects where you need to really work hard in it and spend a lot of time in it to get far. Even the straight A student in GCSE chemistry can get Ds in the first year. It would test your resolve and you would need to really push through to get the grades you want; many have done it before, so it's not impossible.
Maths A Level - considering how much I like maths, I don't care much about your comments. Yes, GCSE further maths do have material from A Level Maths, but again, it's one of those subjects where you need to really work at it in order to get the grades you would need. It's not one of those subjects where you can look at something once and then expect straight As. Pretty much like chemistry, it tends to knock your confidence in the first year, but if you push through you can generally get the grades you want later on.