It's good news bad news, if indeed he is the Father of the child.
The Swiss agency can backdate the claim, and pursue through the UK courts via the
REMO agreement both countries subscribe to. However, the UK courts will not force blood out of a stone, and will take the view that the backdated arrears can be paid off at a rate that will not enforce hardship on your family, current payments, possibly until the child leaves full time education, will be due as per Swiss calculations.
I am of the view that a Swiss court cannot enforce a DNA test on a UK national, and that leaves a huge task for those on the Swiss side to prove who the Father is. Denying being the Father, should it turn out to be true could incur some heavy Swiss fees and charges, but I believe it would be a fair gamble.
There is always the 3rd option of negotiation and settlement.