I am having the same problem - I can't fit both my DS's into the after-school club on the same days (?!), and no childminders in our area do school pick-ups, they only have them until school age. I WOULD be worse off in the short term working, as Nursery fees for DS3 are criminal, but if I can't get ANY childcare for a 9yo and a 7yo, how can I work??
I'm still applying for jobs though, just hoping I don't have to turn any down (ha! likely...NOT), due to lack of childcare.
As for Asda being parent-friendly - HA! again. I have worked for them in the past - I was on a 22.5hr/wk contract, but working 60+ hrs (had to sign opt-out clause for working hours directive or they wouldn't offer the job...). And the Saturday thing pisses me off too - none of the nurseries here do Saturdays, nor do the after-school clubs. So when they say you will have to be available to cover Saturdays - how do you answer when you know DAMN WELL you just CAN'T? I'm an LP with no family to help - I can't leave the dc alone FFS. I can't do short notice, in fact I can only work my contracted hours now, so I'm finding it really hard to get a job.
And that's even without bringing my disability into matters, I won't even talk about that until I'm OFFERED a job now that I don't have to. But it stops me from working nights (too much strain on my system, my Neuro won't sign me as fit for night work OR FT work, only PT work). Who the hell will employ me?
Still looking though. These Zero-hours contracts have become endemic lately, as has employers expecting you to cover sickness at the last minute, or having to work one weekend day when they KNOW there's no childcare. So whilst it's not overtly discriminatory against parents, in all honesty, it IS discriminatory. Good luck proving it though....
Oh, and I'm not allowed to do care work because of the risk of me having a seizure whilst caring for someone vulnerable. Which seems to be all that's available. 
OP - I fully sympathise, if both you and your DP are working FT for min wage, you would have a combined income of only £24K pa BEFORE tax. Does that mean you aren't allowed a second dc just because you're not able to earn a high enough income? Bollox. Sorry, to those of you out there with money, but that's just WRONG. Oh so OP is poor, and TTC. Well she can't, despite the fact her DP works FT, because he doesn't earn enough, and they have to claim TC's. Oh get off your high horse. I can understand that in MY situation, where I have 4 dc already, that if I go back to work then TTC (which I won't do, I'm counting down the 2 yrs till my PCT will agree for me to be sterilised, and not going within 10 feet of a bloke till then!) , THAT would be irresponsible, as I already have more than the 'average' amount of dc. But come on, the OP only has ONE dc. Or are the working poor not allowed to have more two dc if they have to claim ANY benefits? What about all of you on here with 2/3/4 dc that are 'whining' about losing your Child Benefit soon? Should YOU have not had any more than 1 dc because you can't afford it without the ChB? No? Then stop ragging on the OP.
It CAN be almost impossible for someone in OP's situation to go back to college to improve their qualifications - no help with childcare whether you're on benefits OR your DP is working, no help with the course fees either, and If your DP is on a low income (as OP's MUST be to get WTC, he has to be on less than £17.7K pa before tax for a FT job), there IS no money for college. I should know, it's what I was trying to do before Ex-DP left.
Now I'm trying to get a job with no qualifications, having been out of work for over 3 yrs, due to my last place of employment closing down, my disability AND having a baby. An attractive employee? I think not. I probably wouldn't employ me, even WITH all my references stating what a hard worker I am!