I bought a really cheap bass guitar second hand. It works fine, but I'm wondering if it would sound better with more expensive pickups. Since I got it at a market for around £20 I'm guessing that the pickups on it are fairly bad. if I were to invest in new pickups what difference would it make? And would it be worth it? I play and perform every now and then in a jazz band.
It very much depends on the guitar. Do you have any idea what it cost new and how old it is? If you have a new bass guitar for £20, any amount of money you put in it is wasted. It will be CNC-machined from cheap wood, and in few years the neck will be warped enough to render it unplayable.
If it is reasonably old and works fine, chances are that it will stay that way for a reasonable amount of time. "Works fine" means different things to different people: try getting a hold of other people's instruments now and then, just to have some point of comparison.
New pickups will not change the mechanics of the guitar, the precision with which it is fretted, the clearances between string and frets, how well it is in tune (obviously after tuning) and a few other things. But the pickups are a dominant part of the sound character.
Whether they are worth changing depends on what kind of thing you find particularly worthy of improvement with this guitar.
Now the question is why you got the guitar cheap. If you got it cheap because it was in some mouldy basement and is rusty all over, it is worth noting that this does not in any tangible way affect the pickup quality. The pickups don't really get noticeably worse but rather fail completely at one point of time (coil damage). Of course, they may become hard to adjust before then, but the sound itself is pretty much the same.
The pots and switches and connectors are more likely to cause trouble when corroded. Using tuner spray (on the pots) and/or contact spray on switches and connectors may help with that.
I would say yes, but don't expect miracles. A cheap bass Is and always will be a cheap bass. Ofcourse it will Improve the sound but it won't sound like a good bass. If you can't afford a better bass then go right ahead and change the pickups