I can barely get my compost more than 5C above ambient. But I think it is mainly because the compost bin isn't big enough (one of those black plastic daleks), so has too much surface area/volume. Also, it is in a very shady spot.

Shade shouldn't be a problem. In fact, a cover is good because it helps you control the sunlight and rain.

If the pile isn't heating up it's usually either bone dry or there's not enough volume. Aim for at least one cubic meter (or 3x3x3 feet).

If you have fresh grass clippings, add them - they will help a lot. A heap of grass clippings with nothing else will heat up on its own (but don't do it because it will smell).

Your cold compost is going to be fine, it will just take longer and won't cook the weed seeds. If you have more questions there's a composting subreddit: https://www.reddit.com/r/composting/

> But I think it is mainly because the compost bin isn't big enough

Pile size is almost all that matters for how hot it gets, in my experience.