Can a cheap cable prevent AC charging from working? A while ago, I had my phone repaired (the USB socket was worn out) and had to replace the USB cable as well. Since then I haven't been able to quickly charge my phone in AC mode. Previously it worked both on the dedicated charger that came with my phone and the "power" usb socket on my laptop. Is it plausible that this is because of the cheap cable?

My phone is a Sony Xperia Neo V.
It is quite possible that the "cheap" cable is a culprit. You could try a couple of things


Replace the cable with a cable of 24/28 awg thickness. Google for "28/24 awg micro USB" to locate suppliers close by. These cables are thick and offer less resistance to charging current. Alternatively, if you have a spare thick cable of shorter length you could try that and see if the charging is faster.
Download Ampere, It measures the charging current (it does not work on all devices, but your device is NOT mentioned in exclusion list, so it should work). Connect the "cheap cable" on charge and measure the  charging current. Repeat with cable you purchased (24/28 awg) or a spare short thick cable. If the value in second case is higher, the conclusion is obvious. You could even try compare values an  with OEM cable if you can locate with a friend or at a service center. 
You could consider buying an original Sony cable . I found one on eBay costing about $5.