First thing's first: if you study engineering you can still become and accountant when you graduate, however the reverse is not true.
In terms of degrees, engineering is quite a hard degree that involves a fair bit of maths (but not as much as maths or physics) alongside a host of other things, from lab reports to group projects to programming to using other software packages. But it's definitely worth it because it's one of the most versatile degrees out there. I'm not aware of any other degree which keeps as many doors open. I don't know much about accounting degrees, however.
In terms of pay, engineering is generally paid well. You can look up starting salaries for yourself on Unistats and company websites. Of course on the internet a lot of people complain about the salary because they think they're entitled to a lot of money right off the bat, but to actually be a useful employee to the company you need a fair bit of training, which takes time (and money).
Location also plays a role. A lot of engineering jobs are in the somewhat economically depressed parts of the country, where wages aren't generally that high. But even in these locations, some of the engineering graduate wages are very impressive.
You can also work in oil & gas, chemicals, pharmaceuticals, fast moving consumer goods etc. where the pay is generally pretty good. Everyone from my uni goes into oil because it's the dominant industry up here and this year I know of very few people starting on less than £30,000, with some getting as high as £40,000 base starting. So if you're looking for money as an engineer you can definitely find it.
In terms of how easy it is to get a job,
I wrote a post about it a few days ago.