Minimum wage applies to anyone regardless of age. It's set higher as you get older though.
If someone was on £200 p/w this wouldn't necessarily be illegal depending on how many contracted hours they work which for a footballer would be less than a typical 40 hour week I'd guess.
For £200 p/w an under 18 with a minimum wage of £3.87p/h would be able to work 51.6 hours
An under 21 with a minimum wage of £5.30 would be able to work 37.7 hours
For an under 25 with a minimum wage of £6.70 it would be 29.8 hours
For an over 25 on £7.85 it would be 25.4 hours
I don't know how many actual hours a footballer is contracted to work but I suspect that even on £200 p/w only an over 25 would be pushing it. By the time you add in appearance money, win bonuses and any other add-ons its actually not beyond possible they could be on £200 p/w and surely £250 p/w wouldn't be breaking the rules even for an over 25.