If the "Designer" did any original design work, creating a new design for the clothing involved, even if it is a minor variation on a standard pattern, that creates a copyrightable work, whose copyright will have initially vested in the designer. Who provided physical materials is irrelevant. Who provided ideas may not be. If the person mentioned in the question as the "designer" in fact simply implemented other people's ideas and made no original design work, then there may be no copyright. If, however, the designer did original design work to translate general ideas into actual clothing design, then the designer has a copyright, which may or may not be shared with those contributing ideas, depending on the factual details.
Copyrights are not "obtained", in the UK and the US (and all other countries that are members of the Berne convention) they are automatic on creation of any copyrightable work in any tangible form, and a work of clothing design can definitely be original enough to be copyrighted. (But it might not be.)
This will not, however, give the designer ownership of the physical clothes. That is separate from ownership of the copyright, if any, in the design.
As to any images, such as of cartoon characters, used on the clothing, copyrights in those will be that of the original creators of the image. if the designer modified the images, then that probably creates a derivative work, which is a copyright infringement unless permission was obtained or the design had been released under a permissive license.
If these clothes are not being reproduced in quantity for sale, there is a good chance that the copyright owner will not notice, or will not bother to sue for infringement, but that is no guarantee. If the images were used to advertise the show, or are late used to advertise the clothes for sale, and the images are trademarked, that might well be a trademark infringement, and the trademark holder might sue. In the US one may obtain a trademark merely by using a logo or image in trade, but in the UK trademarks must be registered to be protected.
None of the copyright or trademark issues would affect the actual ownership of the physical clothes.
How much design work the designer did and whether a copyright would result is a factual issue that cannot be answered from the info in the question. If the only changes made were to apply the image, then there would be no original design in the clothing itself, and so no copyright in it.
Note that when one purchases a "pattern" for a shirt or dress, even though it is not very different from many other similar articles of clothing, it will carry a copyright notice and be protected by copyright.