I'm not talking proper warm weather plants here, only tomatoes in my temperate climate.
I grow quite a few tomato seedlings for sale, and need to start them early to have them big enough on NZ's traditional planting weekend.
Nights are still pretty cold in early spring here-but we're talking NZ cold, so above freezing.
You shouldn't be too cold in the UK though...
I use a heating pad set around 18 Celsius/65 Fahrenheit to germinate, and as soon as they're up, they go in a cold frame.
In my experience, once a plant's above ground, the most important thing is having it in as much strong, direct light as possible.
Insufficient light and plants are always a bit pathetic, skinny and 'stretched', whereas if they're too cold, growth just slows till they warm.
Tomatoes are special: they grow 'adventitious' roots and it's generally encouraged to transplant them right to their top leaves.
A bit of stretching isn't the end of the world.
So in a really roundabout way, I'm saying if the plants aren't tropicals and there's really good light, I'd say you should be fine