How can I purchase an Apple Watch with monthly payments?

If you are in the U.S. and have an Apple Card or can qualify for one, you can use the interest-free Monthly Installments option when you buy an Apple Watch. See the following.

Products you can buy from Apple with Apple Card Monthly Installments - Apple Support

and

Apple Card - Monthly Installments - Apple

I don't know of any Apple interest-free financing in the UK.

The following has information about financing options they do offer in the UK, but they charge interest. See: Apple Financing with Barclays and PayPal - Apple (UK)