Something quirky to know about the UK Law -in addition to RevengePorn- is that, unlike America, they have Grandparental Rights.
Meaning if you tell his mother? His parents can fight for custody... And in nearly all cases at least partial custody is granted, and sometimes sole custody. Even partial custody often also restricts the child from leaving the country... Especially if he parents claim you're a flight risk... As you are by definition (as you're not a UK citizen). I know a fair number of American moms who've been caught up in this, because it's not a normal thing to even consider over here, where grandparents have no legal rights.
His terror over you telling his mom (and how he links that to child support each time)? Makes me think his parents would immediately file for their grand parental rights, they don't even have to be UK citizens, he is & it's his child... Which would place him squarely on the hook for child support, even if he never saw his child, ever.
Unless you would consider giving up your baby to these people? Or being stuck in the UK for the next 16-18 years, much of that time with no work permit? (also unlike the US, having a British child does not confer the right to citizenship, nor the right to work) Or having to live in a country with no extradition treaty for kidnapping. I know the US doesn't have a treaty with Albania & a few other Balkan states, not sure about the UK. We definitely have a US/UK treaty. Denying a British grandparent there, will get you arrested here. Or anywhere else the UK holds that particular treaty with. Yes it's your child, it's still considered kidnapping if there is court ordered visitation &/or custody.). I would tread very cautiously about whom you tell what to.