Your headteacher's advice sounds reasonable. If you're unsure about studying on a degree for any reason, then an HND might be another option too (From an academic perspective, the content of an HND is roughly equivalent to the first two years of a degree, but also that content also tends to be more vocational in nature. You can usually convert an HND into a degree later on anyway).
As the above poster mentioned, having 2 A-levels certainly won't get you into a prestigious university, although that's not really a problem for a future career in a technical IT/Computing related field.
A-Level Maths puts you in good standing for a lot of Computing-related courses outside of the Russell Group universities. From personal experience, Staffordshire University nearly always make offers to students on Computing-related subjects applying with 2 decent A-Levels - particularly if one of those is Maths (I was in exactly the same position applying to Staffs and ended up being given an unconditional offer - I eventually finished 6th form with A-Level Maths & Computing both at grade B ). Other similar universities might include Bournemouth and Southampton Solent.
Lastly, if you're looking at a career in anything related to computing (Game development in your particular case), then I would strongly recommend you look towards a degree which has a 4-year Sandwich option (i.e. a 3-year degree course with a 1-year industrial placement).
Having a degree is useful, but a 12-month work placement working in the industry that you're looking to move into for your career will put you ahead of the rest of the graduates you're competing against when you finally get out into the real world. (It's also one of the reasons why Universities like Bournemouth and Staffs are actually pretty good for future employment prospects, despite having relatively weaker academic credentials, those universities have decent relationships with businesses and tech companies)