It all depends on this new mapped role you have been offered. If suitable alternative employment is offered, it might be subject to a statutory 4 week trial period if the role, place of work or other terms and conditions are different from the job which you are being made redundant from. Sadly, there is nothing specific which states exactly how much difference there must be between the terms and obviously hey should be substantial, rather than trivial, for this to apply.
If you end up taking the role without the trial period and it turns out unsuitable but the employer refuses to consider redundancy, you may have to look at resigning in stead and claiming constructive dismissal.
Before constructive dismissal is pursued, it is strongly recommended that a formal grievance is raised in order to officially bring the concerns to the employer's attention and give them an opportunity to try and resolve them.
If resignation appears to be the only option going forward, it must be done in response to the alleged breach(es) (i.e. without unreasonable delay after they have occurred, so as not to give the impression that these breaches have been affirmed). Whilst not strictly required, a resignation would normally be with immediate effect and without serving any notice period. The reason is that the whole argument would be that things had become so bad that the employee cannot even continue working there a day longer. It is also advisable to resign in writing, stating the reasons for the resignation and that the whole situation is being treated as constructive dismissal.
Following the resignation, the option of pursuing a claim for constructive dismissal exists. This is only available to employees who have at least 2 years' continuous service with the employer, unless the reasons for resignation are linked to some limited exceptions, such as due to discrimination (on grounds of gender, race, religion, age, disability), or due to another protected act, like raising health and safety concerns or making other protected disclosures. There is a time limit of 3 months from the date of termination of employment to submit a claim in the Employment Tribunal.
It is also worth mentioning that there is a possible alternative solution to this, which could avoid the need for legal action. That is where the employer is approached on a 'without prejudice' basis (i.e. off the record and with protection against these discussions being brought up in future legal proceedings) to try and discuss the possibility of leaving under a settlement agreement. This can be done by asking for a meeting, or it can be done in writing, via letter or email. Under a settlement 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 the Employment Tribunal. However, it is an entirely voluntary process and the employer does not have to participate in such negotiations or agree to anything. There is nothing to lose by approaching this subject with the employer and testing the waters on this possibility - 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.
Please follow this link to ACAS for some more general information about constructive dismissal: