I don't really get this whole "kids must get a job as soon as they turn 16" mentality. I think it's probably a cultural thing in the UK ?
I also don't think it's a bad thing if a child wants to work. But the keyword there is "wants".
Certainly working so young may be beneficial to some children, just as it may not be so great for others. Sweeping generalizations don't work there, it really depends on the kids.
There are also other endeavours that are just as worthwhile as getting a job (learn a language, engage in sports, learn music, whatever).
I think a child develops at their own pace. Obviously there are some basic things they should all do (help with house chores, get decent grades) but I don't thing getting a job is the be-all and end-all. I think the most important thing is fostering a good attitude.
I'd talk to my child if they want more money/seem to be bored and explore options. Getting a job would be one thing they could do but so would taking part in an association, doing sports, doing extra chores for extra pocket money...
If the child is happy, doesn't ask for extra money and keep themselves busy, I don't see the point of making them get a job.
You mention your daughter would like more money, and it's very reasonable to deny her more than a normal amount of pocket money. But if that doesn't motivate her to find a job on her own, forcing her to do it isn't really gonna work. Not least of all, if she doesn't want to be there, people will most likely feel it and that's not going to be a great experience for anyone...
So, long story short, talk to her about it but don't write her CV for her, don't send applications for her, don't hover while she does it (or doesn't), and be ready to accept that she may not be ready to work yet.
Frankly, I don't think time spent working at 16 is valuable. It can become valuable if the child is motivated and turns it into a good experience but otherwise it's just a waste of everyone's time.