Many thanks for your patience. The new employer will have a duty to take you on under your existing terms and conditions and honour them. This means keeping you in the same job and on the same terms, such as hours and pay.
Whilst minor, unavoidable changes may be acceptable, what has happened to you sounds like quite a major one. If your job does not actually exist any more, the employer should be looking at a redundancy procedure. Before they make you redundant, they need to offer you any suitable alternative employment which may exist. What you have been given does not look suitable though so you should be able to reject it and punish for redundancy instead.
Regardless of the above, you cannot force the employer to make you redundant, even if they should. In that case, you will need to consider resigning and making a claim for constructive dismissal in the Employment Tribunal.
It is worth mentioning that there is a possible alternative solution, where the employer is approached on a 'without prejudice' basis (i.e. off the record) to try and discuss the possibility of leaving under a settlement agreement. Under such an agreement, the employee gets compensated for leaving the company with no fuss and in return promises not to make any claims against the employer in the future. It is essentially a clean break, where both parties move on without the need for going to tribunal. However, it is an entirely voluntary process and the employer does not have to participate in such negotiations or agree to anything. It just means that these discussions cannot be brought up in any subsequent tribunal claim and prejudice either party. So there is nothing to lose by raising this possibility with the employer as the worst outcome is they say no, whereas if successful it can mean being allowed to leave in accordance with any pre-agreed terms, such as with compensation and an agreed reference.
Does this answer your query?