The picture in the linked website is very misleading - what you planted are not the sweet peas (an annual) showing in that photo but a perennial commonly referred to as everlasting pea. They are a scrambling plant rather than a vine, have short racemes (don't make a good cut flower) and have no scent.

You can plant them out now but watch out for things like slugs or cutworms.

Yes, GG, I did wonder what I was looking at - not lathyrus odoratus for sure. Had wondered at l.sativus or rotundifolius (I have never grown l.latifolius). But yep, your suggestion seems right to me.

I would plant them out too, with a weather eye on pests. Might try to gently and loosely tie in the slender canes, just to keep them off the soil surface.