A number of commercial cartoonists operate in the US, so you might be sued in US courts (with judgment enforced by your courts), regardless of where you live. Basically, you have to comply with everbody's copyright law, when you operate a web page, unless there's an issue with a strange law in a country that has no legal reach into the UK (no treaty governing enforcement of foreign judgments).
If you knowingly and willfully allow others to post protected material, you can be sued and found liable for major damages. If you unwittingly allow others to post protected material, you can be sued and found liable for lesser damages, unless you've protected yourself by following DMCA safe harbor rules. The gist of those rules is that you have to make it easy for rights-holders to contact you and request removal of infringing material. If you do this in the right way (expeditiously; following a certain protocol for notice and counter-notice), you will not be liable. There are particular requirements on the takedown notice and so on: pushing the matter off onto the uploader will not work. Rather, you must respond to takedown requets. (From a practical POV, putting contributor's email addresses out there in the open is also going to seriously restrict the number of contributors you will get).
Your website presumably operates under someone else's domain, hardware and so on, and that business might be sued as well, for letting you let someone infringe copyright. Undoubtedly they will have included some statement about copyright infringement in their terms of service with you, and they can sue you if you don't do that. It is extremely unlikely that you could hide from a person whose copyright you have infringed (the service provider will comply with requests for information, assuming privacy laws are followed in doing so), so assume that a cease and desist letter can and will be delivered to you, even if you don't provide any contact information.
There is no equivalent safe-harbour law in the UK. This may put some burden on the rights-holder, that they may have to give you "actual knowledge" of infringement (it would not be hard for a rights holder to provide evidence of their rights).
A TOS clause whereby users must vow that they have a right to upload the material does not remove your liability. You could include an indemnification clause requiring an infringer to cover your costs created by his actions, but then you would have to locate and sue the infringer to actually cover those costs. Since the work is being used for profit, you would also be liable for those lost revenues, even if you had paid out a share to the uploader.
Since there is no definitive database where one can determine who holds the rights to a work, it can be difficult to protect oneself against infringers who upload to your website. The risk can be reduced by making it easy for rights-holders to notify you of infringements, and by responding rapidly to such notices. However, it is not automatically safe to take down any alleged infringing materials, because that may infringe on the contractual rights of your contributors, especially since the user provides you something of value in exchange for something that they value (money, in particular). If you just take down somebody's cartoon, you may be liable for lost revenues and even copyright violation. If the uploader had in fact infringed somebody else's copyright, they could not sue you for taking down "their" cartoon: but you don't know if they actually have the rights. To prevent yourself from getting sued by contributors, your TOS would need to include language reserving the right to take down a contribution (this is where you would need to hire an attorney to craft the TOS).
DMCA not only protects a service provider against liability for infringement, it also protects them against customers who want to sue for taking down their material. This page gives an overview of this aspect of DMCA law, and redundantly building such provisions into your TOS is not legally harmful. Unfortunately, one of the provisions is that you "not receive a financial benefit directly attributable to the infringing activity", which would not be the case if you are making money from ads where cartoons attract customers.