I currently work a single full time job, although I am looking at redundancy in the next 4-8 weeks. I have secured a new job, but they require me to start in the next 4 weeks. If I take the new job I will lose my severance. I am thinking about working both jobs in order that I don't lose my severance. This is something I shouldn't do (as both companies are competitors), but is there anyway either employer could find out (through tax or background checks, etc)? It would only be for 2-4 weeks. Thanks.

There is no law that says you can't work 2 jobs at once evn if they are competitors. However, did you old job require you to agree to a non-competition clause?