What can be in an unheated greenhouse?

I've never had a greenhouse before and don't really know what can be out there at different times of the year. However, my dining room table (by south facing french doors) can't take any more seedlings so I could really do with moving things either to the greenhouse or outside as soon as possible. My dining room has the heating turned off, but does get warm during the day because of the big windows.

I have:

Onions (Red Baron, Ailsa Craig, Welsh Bunching)
Leeks (not many - had rubbish germination)
Cabbages
Lettuce/salad mixes (have a container of salad in the greenhouse now, but that was started out there)
Tomatoes
Sweetpeas 
Cauliflowers
Artichokes
Allium (yes, singular - I didn't do well with these!)
Zinnias
Tomatillos
Cucumber
Watermelon
Lemongrass
Radishes
Mustard
Kohlrabi
Chillies
Aubergines

I'm thinking definitely not the chillies, tomatoes, cucumbers, watermelons, or aubergines.

Moving the onions would free up the most space. What do you all think?

Think of Mediterranean veg - they need it warmer overnight. Cucurbits like it hot.
Think UK veg, = they're hardier. If you think its going to be really cold overnight,. cover them with fleece or some polythene.

So how worried should I be about my hardier seedlings overheating during the day?

What a lot you’ve got. Lucky you.
My zinnia are not sprouting at all.

So how worried should I be about my hardier seedlings overheating during the day?

Do you have an automatic window opener on the GH? Its worth getting one if you haven't. If its forecast to be hot, leave the door/window open.

So how worried should I be about my hardier seedlings overheating during the day?

Do you have an automatic window opener on the GH? Its worth getting one if you haven't. If its forecast to be hot, leave the door/window open.

An automatic opener is on my long long list of things to get to.

Knowing me, the chances that I will go up the garden to open the greenhouse door before I dash to the bus-stop to go to work are pretty low! Not a morning person   So an automatic opener is definitely a must. At some point.

What a lot you’ve got. Lucky you.
My zinnia are not sprouting at all.

The ones I bought from Lidl are doing really well, but I've not had any luck with the Orange King seeds from the swap just yet. Hoping they just need a little bit longer, but I may have kept them a bit wet. I may try sowing some more if there's nothing doing in the next few days.

Ideally you would put them on staging in the greenhouse as, on the floor, slugs can be a big problem with little seedlings. I've got onions. leeks, brassicas, beetroots and lettuce out.we've had several frosts and I've accidentally left the door open a few nights and they are all coming along nicely.
The onions and leek are only like blades of grass in seed trays but I'd like to get them planted out. I did this last year and got a bumper crop of both with not too much effort (sow the seeds, plant out, hoe a couple of times then harvest).

To answer your question directly, this is what I'd do:
Onions (Red Baron, Ailsa Craig, Welsh Bunching) - greenhouse or outside
Leeks (not many - had rubbish germination) - greenhouse or outside
Cabbages - greenhouse
Lettuce/salad mixes (have a container of salad in the greenhouse now, but that was started out there) - greenhouse
Tomatoes - indoors
Sweetpeas - no idea
Cauliflowers - greenhouse
Artichokes - greenhouse
Allium (yes, singular - I didn't do well with these!) - greenhouse or outdoors
Zinnias - no idea, but I'm guessing greenhouse
Tomatillos - indoors
Cucumber - indoors definitely
Watermelon - indoors definitely
Lemongrass - indoors definitely
Radishes - out!
Mustard - greenhouse
Kohlrabi - out
Chillies - indoors definitely
Aubergines - indoors definitely

I haven't got all of these types sowed yet, and I've not grown caulis before, and I am by no means an expert so do tell me if you disagree

What a lot you’ve got. Lucky you.
My zinnia are not sprouting at all.

The ones I bought from Lidl are doing really well, but I've not had any luck with the Orange King seeds from the swap just yet. Hoping they just need a little bit longer, but I may have kept them a bit wet. I may try sowing some more if there's nothing doing in the next few days.

Don't blame yourself, it's more likely that I didn't let the flower heads dry out enough or the seed didn't develop properly. Gosh they were a real faff to extract the seed from!