In Honolulu, basically every bar or club closes around 4am in Waikiki, and around 2am downtown. Most places do not really have a cover, but if they do it wont really be too much more than $10. London is much different. All of the bars I have found so far close by 11 on weekdays and midnight on weekends. Then, if you want to stay out, your only option is to head to a club. Some of the clubs will stay open until 4 or 5, but you will have to pay a pretty steep cover charge. Most are between 10 and 20 pounds, aka, 16-32 US dollars.
London is definitely an extremely fun city with a very active nightlife, however, it is worth noting that it can get expensive quick.