The law conversion course is the graduate diploma in law (GDL). It covers the 7 core modules that a qualifying law degree contains which are land, equity, criminal, tort, public law, EU law and contract. After the GDL you would then have to do the LPC or the BPTC depending on whether you want to be a solicitor or barrister.
Generally, a non law graduate is not at a disadvantage compared to a law graduate. Most firms have a relatively even split between the two.
If you're in year 12 though and looking at becoming a solicitor, the process may well change though. The SRA are looking at implementing what's become known as a "super exam". It's difficult to say how different it would be if implemented, but it's something worth being aware of if that's the career you want to pursue.