Thank you. The relevant law on associated employers is here:
You will see that as long as both are companies of which a third person (directly or indirectly) has control, then they can be construed to be associated employers, in which case your employment with both of them (assuming no break in between), will be continuous.
You can raise this argument with them first but if they refuse to accept it, your only option is to make a claim in the employment tribunal.
Before a person can make a claim in the employment tribunal, they would be required to participate in mandatory early conciliation through the Advisory Conciliation and Arbitration Service (ACAS).
The purpose of this process is to allow ACAS to mediate between the claimant and respondent to agree on an out of court settlement in order to avoid the need for legal action in tribunal. The respondent does not have to engage in these discussions, or if they do and the talks are unsuccessful, the claimant will be issued with a certificate allowing them to make a claim.
However, if a settlement is reached, the claimant would agree not to proceed with the claim in return for the agreed financial settlement. Other terms can also be agreed as part of the settlement, such as an agreed reference.
To initiate the conciliation procedure ACAS can be contacted online by filling in the following form (https://ec.acas.org.uk/Submission/SingleClaimantPage), or by phone on 0300(###) ###-####