The 'complete stock market' is immense. Do you mean just the FTSE-100, FTSE-250, AIM, PLUSMarkets?
The London Stock Exchange Site allows you to create your own 'watchlist' of up to 40 companies, but you can create 5, 10, 20 different watchlists so you could have your own FTSE-100, albeit on 3 pages.
Thisismoney.co.uk also has a 'portfolio' option - similar to a watchlist but designed for shares you actually own; I use it with nominal holdings of 100 shares in each company on the portfolio. By including the nominal purchase price you can use it to have a shadow portfolio and see how your investment decisions would have worked out.... it can be quite educational.
Other providers you might look at are ADVFN and Hemscott, although I'm not sure exactly how they work; they also send e-mail alerts and am/pm market round-ups.
All of these sites supply data that is at least 15 minutes old - if you want real-time information you will have to sign up to a paid-for service.
Hope this helps.