I think that some of the people who sell Birdy bikes might have some spare parts although not listed on the internet. However, have you tried servicing the freehub? See, for example, and . If that doesn't work then ,if the rim is in good condition, one option is to build a new wheel using the old rim and a new hub. From what I've ready the Silent Clutch freewheel mechanism has been replaced by the standard (clicky) pawl and ratchet freewheel mechanism. What really matters is the distance between the drop-outs. Any hub of the right dimensions will fit the bike and this opens up the opportunity to fit, for example, a 9 speed cassette with a wider gearing range although a new derailleur and shifter will probably be required.

You've therefore got several options: (i) Attempt to service the freehub. (ii) Rebuild the back wheel yourself while keeping the 7 speed gearing;(iii) Rebuild the back wheel using a hub that accommodates a wider range cassette; (iv) Contact a business with known skills in maintaining bikes, preferably Birdys to see if they will do the work. A couple of possibilities are:
https://www.kinetics-online.co.uk/about-us/
https://www.avonvalleycyclery.co.uk/