Try and figure out exactly what you need as 'quite a big van' covers a fair range of vehicles and prices so one price for 'quite a big van' isn't really going to be comparable to another.
transits and sprinters come in three lengths, but not everybody stocks the largest size, so when they say 'long wheel base' sometimes they are referring to the middle size van. Vivaros are smaller than transits and "long wheel base' vivaro is roughly comparable with the smallest size transit.
If you can get a SWB transit for about £30 thats an OK price. That gets you a load area 8ft long and about 4ft between the wheel arches and about 4ft tall
if you can get a MWB for £35/£40 thats an OK price. 2ft longer, perhaps a foot taller (but some will call this a LWB)
if you can get LWB for £60 thats and OK price. about 12ft long. Some will call this extra long wheel base, or a Transit Jumbo
Luton Vans are around £60-£80 About 7ft wide, 7ft tall, length can vary from 10 to 14ft though, so worth asking for measurements to make a comparison. Lutons are thirsty, if the hire company's stock isn't very new then lutons can also be leaky and not very secure. Its also very difficult to make the best of a luton if you can't secure large/ungainly items well. A good luton will have tie bars along the wall that you can rope your furninture to, a bad one will have nowt and the interior, the roller shutter (and consiquently all your belongings) will be smashed to ****
Important factor with lutons is although they are cavernous they don't carry a great deal of weight and are very easy to overload. If you are going to have to do more than one trip use a luton to move the big and awkward stuff (beds, sofas, anglar awkward items) then do the rest of the boxed heavy stuff in a smaller, cheaper van. A MWB transit can carry nearly twice as much weight as a big luton.
Mark out an area 8ft x 4ft on the floor and imagine how your stuff will fit in it, then mark one out 10 x 7ft. That should help you work out the right tool for the job.
Get prices, but try and give the stock a look over once you find something cheap – there are some shocking vehicles out there.
Push to get all the pricing info over the phone, don't take the cheapest quote only to find the add on damage waivers, milage prices, fuel and all sorts of other guff when you collect
For house moves keep in mind that left unattended in a van your belongings aren't likely to be insured by your home contents, especially not if you leave the van unattended and pretty much never overnight. Some will will cover 'in transit' but thats literally while the vehicle is on the move.