I'm not a plumber but...

Generally you need two things:

1) Floor has to be rigid or tiles will crack. On a wood floor this normally means some form of boarding has to be laid over the whole floor to stiffen it and provide a good base for tiling.

2) The walls and floor in and around the shower need to be waterproof.

Both can be achieved by boarding out with something like wedi board  or hardibacker board. These are also used on walls in preference to regular plasterboard.

If you were having a totally open shower in middle of room then tanking the whole room would be the way to go. If the shower is in a corner and there will be a shower screen on one side then you can get away with using wedi board on the floor and just the two walls that form the shower...perhaps there is already something behind the tiles on the walls?

There are several makes of shower tray kits designed for wet rooms. These go under the area where the shower and drain will be to form a slight slope towards the drain. Wedi make several to compliment their standard board.. See links bottom left of here..

http://www.wedi.co.uk/fplumb.php

I would make the side screen taller than 1500mm. Go for about 1800-1900mm tall as much less water will escape over the top.

If you didn't want to tile the shower itself an alternative would be something like one of these low profile trays let into the floor...

http://www.bathroomheaven.com/sp_brochure/stone_resin_trays

..but they are best used with a door as well.

A family member has a mosaic tiled shower and recommends againt mosaic. Says the grout lines are impossible to keep clean and the large number of them makes the whole shower look grey/dirty after awhile.

I'm afraid wet rooms are not a budget job.  The floor area needs sealing, be it fiberglass or purpose made membranes, these have to extend up the wall too.  The floor gully needs to have a fall towards it so floor structure will require some work to allow this.  Then theres the cost of the floor tiles themselves.  Tile grout is not waterproof if water pools on it, the water will get through.

£1500 sounds woefully short.

I do laugh when customers dictate prices to trades, I think I will do that next time I go to the supermarket (I wonder how far I would get?).
Day rates vary depending on where you are in the country and mainly vary by the length of time you will be on a job. The longer you are going to be on a site the less your day rate. You will struggle to find somebody around here who will charge £150/day for a 3 day job, maybe if the job lasted over a week. Then you get London where I know labourers who clear £100 a day after tax.
Are you sure the tiling is 20 square feet, you must have a tiny room? My corner bath has a footprint of about 12 square feet and that does not include any walls?

I would be weary of painting things on, you can't beat a waterproof membrane.Most of the membrane water proofing systems use a paint on product, then a membrane goes on top of the painted on product (some will also recommend a further coat of the paint on product over the top of the whole lot).
Wedi is a rigid board product, which acts as a board, membrane and insulation. Google Wedi and you will get an idea of what they make and how it works, while you are at it Google the stuff that your tiler said he is going to use and see if he is following the manufacturers instructions (and not only using half of the "system"). There are similar products to Wedi or that will give the same result as Wedi but the big difference in price is normally down to where you buy it from and not the actual product. I have bought Wedi for less than any of the other products on the market (at the time), even though the RRP of Wedi is normally higher. Shop around for the best deals.

If you take the tiler @ £450 (I am guessing this is his labour only price),
The plumber at @ £250
The tiles (£30 - £300/square meter), adhesive, spacers, grout etc,
The plumbing fittings, pipes and fixtures (roughly £300 for a thermostatic shower mixer),
Paint and silicones etc,
The waterproofing system for tanking the room,
The cost of bagging and dumping waste (not so much of an issue if you are not trade).
The driving around to collect materials and numerous phone calls to suppliers etc.

It soon adds up!