Being bothered by being in pull-ups is a good thing as it means he will be more motivated to try and be dry! I run an enuresis clinic and find that where the child isn't bothered, and that they are there because it's the parent that is more bothered, actually means we don't often get very far because the child has no motivation to be dry. I don't personally believe in reward charts (you are rewarding a child for something they have little or no control over) but offering incentives can't hurt. Don't put too much emphasis on 'achieving the prize' or making it conditional on being dry as this can add pressure which in itself causes anxiety which can cause wetting. Reward a behaviour ie drinking more fluids, going to the loo more regularly, getting up to change PJs if wet or coming to tell you rather than lying in the wet bed, rather than rewarding the outcome ie a dry bed. This way you are rewarding something the child can actually influence rather than wetting, which in most cases is beyond their control.
There tends to be a few possible reasons for the nightime wetting - a low level of a hormone we all start to make from the age of about 3 can mean that the kidneys make as much wee at night as they do in the day, so the bladder can't hold it all and the child wets. You can get a tablet form of this hormone from the GP/paediatrician (don't use the spray, it's awful having something squirted up your nose!) - the GP can prescribe DesmoMelt which is a melt tablet so just dissolves under the tongue. This is usually indicated where the child is still producing gallons of wee, often early on in the evening - the parent will check the child before they go to bed and find that s/he has wet.
Another reason for night time wetting is a lack of arousal - this is when the child just doesn't wake when they wee, even when they are soaking wet. Pull-ups don't help this as they keep the child so dry and comfy, so they don't get a message to the brain saying oy, wake up, you're wet. A way to try and increase arousability is to remove the pull-ups, protect the bed with waterproof stuff and see if they start waking when they wet the bed. If they don't this is when you can try an alarm. These can be body worn, or a mat, which go off (loudly!) when the child wets. The idea is that the child gets up, goes to the loo to finish weeing and then changes the bed and PJs. After a while the brain gets a stronger signal that they need to wee and starts waking them when they need to go, not that they have already been. A bit like you waking up every day a few minutes before your alarm goes off.
Lifting doesn't help as it reinforces the 'wee whilst asleep' message to the brain, as children rarely wake up properly when they are lifted.
The most important thing in helping night time dryness is bladder training; the bladder is a muscly bag and like any muscle gets stronger if given a workout. A strong toned bladder can hold more wee than a weak flabby one, and so can usually hold enough wee to last overnight or send a stronger message to the brain to wake up when full. To train the bladder and give it a workout it needs filling and emptying regularly, so encourage 6-8 good sized drinks a day (ie about 250ml) and aim to have most fluids during the core of the day so from breakfast time to dinner time, tailing off and stopping an hour before bedtime. The child should aim to wee every 2-3 hours and not hold on until bursting. This regular filling and emptying strengthens and tones the muscles.
Avoid red and brown drinks, so coke, hot chocolate, berry drinks (Ribena, apple and blackcurrant etc) and tea and coffee; they are all diuretics (make you wee) and in many children blackcurrant irritates the bladder.
A good website is Eric.
Good luck