I'm sure they'll be some gardeners along shortly but, for now, I'd just say that using a paraffin heater with a 'home made propagator' could be a very dangerous thing to do if, as it sounds, you're talking about having it in the caravan. Incomplete combustion with paraffin heaters can easily cause deadly carbon monoxide to be given off.

You haven't said precisely which veggies you want to grow, but in the circumstances, it'd would surely be more sensible to cover the planting area with black plastic or something to make the ground warmer earlier in the season, and buy ready grown seedlings or small plants and grow on from there, outdoors. You might need row covers or cloches to keep them warmer if you plant earlier, depending what you're growing - you're in a mild part of the world anyway though. Doesn't sound in the least a good idea to start seeds off in your static home, not least because of damp and the increased risk of fire hazard - you likely will have to just wait till your home is built and you've moved in to grow from seed indoors again. That or get a greenhouse or a polytunnel.

you could use a temporary coldframe made from reclaimed timber and old windows, packed with polystyrene to help insulate them. Any thing on offer on freecycle type sites?

but warming the soil first as bamboo suggests is a good idea.

many crops beans/cabbages, carrots, parsnips etc can be sown directly outside in your prepared beds. check and follow the instructions on the packet.

2l pop bottles make handy mini cloches, cut off the bottom and then you can take the top off to allow ventilation.

welcome to GoY too :o)

A relatively inexpensive and easy to dismantle greenhouse can be built with schedule 40 PVC pipe, plastic fittings, and plastic sheeting. Polytunnels set up inside of that will be warm enough to start seeds of cool season crops now, and will be warm enough to start early crops of warm season veggies later.

Have you, or are you planning to, prepare the ground of your new garden now? Just wondering if that might be a good priority. You can then grow potatoes which are a recommended crop for new ground, and will grow without chitting first. Peas, broad beans and salads can be sown in situ. Perhaps best to forget about raising your own plants just for this year. Or when the weather warms up a bit you might buy a moveable cold frame where you could raise a few hardy plants to get you started.

Get a soil thermometer. When the soil is 39 deg.C. its the time to sow seeds. Or the leaf buds on certain trees.
I once read about a farmer, who tested whether it was time to sow wheat by taking his trousers and pants down and sitting on the ground. He could feel if it was warm enough !

A few days for the wet seed to swell can also be allowed.
Also sow half the packet, then the other half 2 weeks later
on. When buying in vegetable plants from garden centres always harden them off, they are usually a bit weak and
spindly. This is where a cold frame comes in handy.

What about asking around your new neighborhood, you may have someone who has a greenhouse you could share, or give you,another option maybe for this year only concidering your doing the house up would be to buy some starter veg plugs around April/May time would save you a lot of messing about.

Yes, my first thought was also to buy the plants this year and concentrate on getting yourselves more comfortable and the garden more organized for next year. It sounds as if you have your work cut out! Good luck with it all!