For your budget, I'd definitely consider second-hand. £150 for something new is simply going to buy fancier versions of the same low resolution 800x480 pixel-based projector you first had your eye on. It's money down the drain, IMO.
In the used market your choices will be between native 720p and native 1080p projectors. Have a look at the member's market at AVForums. Here are a few that fit your budget:
https://www.avforums.com/threads/sony-vpl-hs10-317-lamp-hours.2233393/
This is a Sony WXGA 1280x800 three chip LCD projector, so it's classed as a 720p unit. It has low hours on the installed lamp and comes with a spare lamp too. The price is £150.
Pros: Low running cost thanks to the low hours and spare lamp. 3-chip LCD means no rainbow effect
Cons: Contrast is low by more modern standards. Limited to a maximum of 1080i resolution; that's fine for Sky/ Virgin/ streamers/ BD players but may be trickier with some consoles
https://www.avforums.com/threads/panasonic-pt-ax200-spare-lamp-£150.2220232/
This Panasonic is another 3-chip LCD unit, but newer than the Sony. It's a true native resolution 720p unit. £175 with a spare (compatible) lamp.
Pros: Later tech, better contrast, higher brightness. Features lens shift which is incredibly useful for getting the picture lined up on screen without having to resort to electronic keystone correction.
Cons: Not a lot to complain at here for the money.
https://www.avforums.com/threads/optoma-hd-131xe-1080p-full-hd-3d-projector-£135-sold.2236414/ (SOLD @ £135)
Single chip DLP projector. Native 1080p. 2000 ANSI.
Pros: Brightest projector of the bunch here, and with the highest contrast ratio.
Cons: No lens shift.
The closer we get to this year the lower the price of budget projectors new. The Sony was £2000 when launched back in 2003, and that was considered cheap at the time. The Panasonic was around £900 new when launched in 2009. The spec was very good as there was a lot of competition around the £1000 mark from Sanyo, Hitachi, Optoma and others at 720p resolution. On the horizon though were the first of the native 1080p projectors, albeit at higher prices. The choice then was pay extra but sacrifice features such as lens shift, or stick with 720p and keep all the bells and whistles.
The Optoma launched in 2013 with a native 1080p panel @ £800. It was classed as an entertainment projector rather than being a home cinema projector like its bigger sibling the HD30. The difference between the two classes is that entertainment projectors are generally brighter but with lower contrast. Home cinema machines sacrifice brightness but gain better black level and improved contrast as a result. The HD30 was £1100.
Buying from the enthusiasts at AV Forums will generally mean you're dealing with someone who has looked after their gear. Ebay is more of a shot in the dark.
£50 for a 16:9 screen..... Nah.
Unless you drop incredibly lucky or you have low expectations, then even second-hand you're probably going to end up with someone's cast-off budget manual or cheap electric screen complete with poor light reflection and surface ripples.
You can see from the images in
this listing* what ripples look like.
This
Beamax 6ft frame screen* is excellent for the money, but may not be the most practical solution. This
Beamax manual pulldown screen still has ripples but from the picture they don't look too awful. Beamax was a decent brand. Sadly the company went bump several years ago.
Personally, I'd steer clear of Panoview and Optoma screens (
Panoview by another name). Lots were sold as cheap
get-you-going package deals with Optoma projectors. I've installed a fair few and even from new they rippled. They're the same as most of the Chinese low cost pulldown screens.
[* Ebay listings, so the links will only work while the listings are live]