You would be held primarily liable as you failed to accord presedence on the major carriageway I am afraid.
It is not for the other side to prove how slow or fast they were going, but for you to prove that the other driver was drving at a speed inappropriate for the conditions, and unless you have a Police investigation, or an independent accident investigation there is little prospect of that happening.
Although it may seem unfair, (and to be fair I have not seen the evidence) this sounds like one of those crashes you may just have to take on the chin I am afraid