Your employer should have provided you with a 'written statement of employment particulars' within two months of commencing work:
https://www.gov.uk/employment-contracts-and-conditions/written-statement-of-employment-particulars
However if the letter you received (offering you the job) and the staff handbook jointly include all of the relevant information then there was probably no need for your employer to issue a separate document.
Even so, it might be worth showing them my link and asking for that 'written statement'.
Being allocated to working with a different department of the company is unlikely to be regarded as a 'change of contract' unless either:
(a) that department is located a significant distance from the original one ; or
(b) the nature of the work is markedly different and outside the scope of what you might have been expected to do originally.
(NB: Since your job title was given as 'general assistant' within the delivery department you couldn't complain if, say, you were taken off driving for that department and reallocated to filing their invoices. So its unlikely that you'd have any right to complain if, say, you were allocated to filing invoices for the neighbouring 'goods inward' department).