Hi Floss,
Much depends on why you want to do this. If it's as a parody or more general humour (for instance as a meme) you will probably not face any issues. However if you want to sell anything which incorporates the altered logos or use the altered logos in advertising or packaging to sell something, then you can expect pushback, especially from big brands like Nike who defend their intellectual property rights robustly.
The main purpose of trade mark law is so that a consumer can rely on a trade mark as a guarantee of a certain level of quality, and producing lookalikes which confuse people about the origin of the goods tends to be treated as infringement. The brand owners are also keen to avoid any dilution of their brand identity to the extent that their trade marks get used generically (for instance referring to all makes of vacuum cleaners as Hoovers). However they can only take legal action if the mark or logo complained about is being used in the course of trade. Mere humour or parody is not sufficient.
From a copyright perspective, problems are only likely to arise if the original work is an artistic work (many logos won't meet this threshold) or you decide to use more than a few words of text or a jingle (for instance) from the original and only change a word/note or two. If your intention is to parody the product or company you should be covered by the fair dealing exception to be found in
section 30A of the Copyright Designs and Patents Act 1988. We don't yet have full idea of what the British courts would consider to be parody, pastiche or caricature.