It's not really designed for this, but, if you insert the following immediately before the end of the PROCcheckclip routine...
3081 : IF text_available% THEN PROCsaveTXT("<Wimp$ScrapDir>.winclip"):SYS "ClipboardHolder_Copy",1,&FFF,"<Wimp$ScrapDir>.winclip"
... then every time that it checks the clipboard and finds that it contains text, it will be copied to the RISC OS clipboard.
Obviously this is terribly inefficient, and incorrect. WinClip would, for this, need to check whether the contents had changed - I would assume there's a notification that goes around that could be used. If not then a length check + simple CRC of the contents would suffice, I imagine.
Handling of other formats is left as an exercise for the reader. Sadly, my laptop is currently on vacation so I can neither try this nor say if it's even correct.