Figured I might as well chime in, for what it's worth... I've owned my Warwick for going on 8 years now, gigged, recorded and toured with it heavily so I might have something to say. It's a 1999 model 6 BO with SD Basslines in it, Ovangkol neck & body with Wenge fretboard.
The NT models have a shorter body. I've not tried one out but I suspect they have a tendency to neck-dive.
My bass doesn't neck dive, but it's shape is definitely different to play than other basses. It does tend to be a bit more 'horizontal' than other instruments and the neck is sort of more 'left' on the body than other basses - the neck sticks out more in relation to the body, is what I'm getting at, so if you have short arms it may be a struggle. It's definitely not for everybody. One of the reasons I like it is because it's so different - it really is unlike anything else out there, by any brand. I haven't, to the best of my knowledge, seen another identical or even mostly similar bass from another manufacturer anywhere (if there are, let me know!)
For me, the Thumb is *the* Warwick sound - that tight, growly, low-mid that cuts through anything. No other bass in their range really quite epitomizes it like the Thumb. It's not the most versatile bass around for sure - it's a one trick pony, but it's an extremely good trick. It's heavy, but the dense, dark wood is what gives that glorious sound. FWIW it's identical in weight to my Peavey Grind 6.
Weight-wise I can get through gigs easily enough, especially with a good comfy strap. I think a good strap is really underrated - I tried lots of them before I settled on an awesome Minotaur strap (check 'em out). Definitely don't scrimp on the strap - it's what holds your (potentially extremely expensive) bass up and makes it comfortable for you to play on.
So to answer the question, it's not neck-heavy, it just has a unique shape that needs getting used to. Once you do get used to it though, it's fantastic. To record our latest album, I tried a Fender Jazz Deluxe, a Pedulla and my Peavey - the Warwick blew them all away for what we do.
I haven't had much experience with newer models, from what I gather things have changed, but the Thumb is my main bass and it's a permanent keeper. Other ones may come and go but this one is here to stay.