Thank you for the clarification sally.
A few significant points:
"I have no fixed term contract and they can terminate my time there with one week's notice, and vice versa. "
you have worked for them for more than two years therefore they must have a valid reason ( your job is redundant etc) to terminate your Contract.
"I was told that my actual work will be absorbed into the duties of existing staff,. "
they are doing away with your job, this will be easy for them to prove at a tribunal
"but found out yesterday that a new person has been employed with a similar job title to mine"
gossip, it may be similar TITLE but not EXACTLY your job.
Legally you can take them to a tribunal, whether you win or not is not for me or anyone on here to say, except it will cost a lot of money take a lot of time and not look very good on your CV.
If I were you I would keep on very good terms with your former Employer ans start looking for a job asap.