Oh god, trust me, getting two songs in sync is a massive nightmare. Sure, there are tools like Sony ACID Music Studio that can do it for you, but it's still hard and I always found it better to use Audacity. It takes even longer to do such mashup than a brand new song sometimes.
The problem with non-electronic music played only with real instruments is that the band might not perfectly play at the same tempo during the entire song and just a 0.5 BPM difference can make it a chore to do any mashup, so inevitably you need to do some editing.
That said, I challenge you to make a song mashup without using any DJ/computer equipment
(I did before)