NSAIDs non-steroidal anti-inflammatory drugs have the potential to raise blood pressure too, as do oral steroids, but you're not likely to get much of an effect on your BP from locally injected steroids.
As for NHS v private steroids (Bonzo) - it might be the difference between using generic versions and branded ones, but who's to say that the NHS always use generics, and the private sector always use branded drugs? The latter will charge more but they won't tell you which manufacturer's drug they are using: they might well still be using cheaper generics!
And within general practice there ARE specialists: in my practice each partner had a special interest and expertise in certain ailments. One partner did all the minor surgery and joint injections, another did all the ante-natals, another all the diabetic follow-up, another all the cardiovascular stuff, another the respiratory problems, etc, etc, so we could internally refer people within the practice to the most appropriate practitioner.