This is why people shouldn’t seek employment advice in AIBU.
https://maternityaction.org.uk/advice/rights-for-parents-with-more-than-one-job/
Can I work for a new employer if I am receiving SMP from my old employer?
Before the birth: you can start work for a new employer and still receive SMP from your old employer before the birth (regardless of whether you were employed by the new employer in the 15th week before baby was due).
After the birth: once your baby has been born you cannot continue to get SMP from your old employer if you do some work for a new employer, unless you were employed by the new employer in the 15th week before your baby was due.
If you do some paid work after the birth for a new employer who did not employ you in the 15th week before your baby was due, you must tell your old employer to stop paying your SMP. If you were paid your SMP in a lump sum, you must return any overpayment to your old employer.
If you go on maternity leave with your new employer then you can continue to get SMP from your old employer.
Once your SMP period has ended (with your old employer), you can work for your new employer or work up to ten keeping-in-touch days during the remainder of your unpaid maternity leave with your new employer.
You can find Government guidance on working during your SMP period here if you need to show it to your employer:
3.12 Working in your Maternity Pay Period
If you are getting SMP from one employer and, before your baby is born, you do some work for another employer, your SMP is not affected.
Your SMP will stop if after the baby is born, but before the end of the Maternity Pay Period, you work for an employer who did not employ you in the qualifying week. It is your responsibility to tell the employer paying you SMP about your new job. You must do this as soon as possible, and make sure you return any SMP payment you get that covers the week you started work and any part of the period after you resumed work.
If you do any work in a self-employed capacity during your MPP, then such work will not affect your SMP.
You can find this guidance online, see section 3.12: https://www.gov.uk/government/publications/maternity-benefits-technical-guidance/maternity-benefits-technical-guidance