St John Ambulance do some excellent courses. You can get the basics on a one-day course, or sign up for various longer courses.
If you have one, see if your employers needs/wants/can be bullied into training more first aiders (i.e. you). A First Aid at Work course takes about a week and leaves you fairly well prepared for most scenarios. If your employer has an automatic defibrillator, get trained on that too. (The main value of this is that you have to be refreshed in resuscitation every six months - that really makes it stick in your mind!)
Where I work we also do "scenario" training. On the training day, you work as normal, but are called to deal with a string of emergencies. You don't know what they are until you get there - and I can tell you some of the "victims'" acting is pretty convincing. There's gallons of fake blood (at least, I think it's fake) and they get someone who knows what they're doing to do make up and special effects. Afterwards, you're shaking!
But you learn from the experience. The most important thing, often, is simply to take control and be bossy. There's usually some bystanders (who don't always know it's an exercise) faffing about, arguing over what should be done. At least at work, as the First Aider (with a big green box), you can step in and say You do this, You run and do that, You hold this... and order emerges from the chaos. There's a sense of relief that someone's taken responsibility, and it's not them!
Anyway, returning to the question - if your employer won't pay, get yourself to St John's. Often they team up with local adult eductation organisations, so you can do it that way. Hopefully, you'll never need that training, but it's good insurance.