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You could replace any/all of the parts with stacked ply. It's just a glulam beam at the end of the day. No need to wait for glue to dry, the screws should hold it all till everything dries.
Leaving centre boards long would even allow for mortice tenon construction. If you then leave a corresponding gap in the joining board. (Or lap joints, or finger joints etc...)
Chem bolts into the wall would be nice and secure.
You pays your money and takes your choice with any of these options. Time, cost, ease etc...
Cheers. Just to check, you mean replacing the wonky uprights with stacked ply? My only concern with that is the extra time required. As it is cleaning and cutting the uprights to length is a small standalone job I can fit in. Measuring, going to B&Q, screwing and gluing, then fixing is four standalone jobs. That said it would look nicer.
Thanks for the plane(?) suggestion.
And @Nef thanks for the reassurances about fixing it.