Here's some advice I gave to my brother recently for which budget laptop to buy.
I've checked Dell, and configured two to what you'd need: (just using Intel 32-bit CPUs)
http://configure.euro.dell.com/dellstore/config.aspx?c=uk&cs=ukdhs1&kc=305&l=en&oc=N06605b&s=dhs&sbc=inspn_6000 The Inspiron 6000 comes out at ?815. Mobile CPU (not great), 1-year collect/return, 768MB, 40GB, 15.4" screen, DVD RW
http://configure.euro.dell.com/dellstore/config.aspx?c=uk&cs=ukdhs1&kc=305&l=en&oc=N06935&s=dhs&sbc=inspn_9300 The Inspiron 9300 is ?1065. Mobile CPU (not great), 1-year collect/return, 512MB, 40GB, 17", DVD RW
http://wehavethings.com/uk/product.php?xProd=56462&froogle No idea what wehavethings is like, but Fujitsu is good. 3GHz Pentium 4, 512MB, 80GB, 15.4" widescreen.
As for Athlon-64...
http://wehavethings.com/uk/product.php?xProd=61459 This one (?740) also has svideo out (which your telly should have). Widescreen 15.4"
Full spec here: http://www.fujitsu-siemens.co.uk/Resources/211/38997532.pdf
Also available at http://www.laptopsdirect.co.uk/asp/product.asp?recorprod=1&product=296&cat=80
Acer do athlon64 laptops, but I wouldn't recomment them cos of the support trouble I had - waiting 6 weeks for it to be sorted.
I'd be tempted by the fujitsu, but you may prefer a 17" screen - the Inspiron 9300 (but with a worse cpu). If you go for an athlon64 computer, check first if they'll bundle XP64 Professional (the only version of XP that's 64-bit). 32-bit XP will run fine, if you're happy upgrading later.
Generally, try to avoid mobile cpus (ie, centrino) as you just want lots of cpu, ie, athlon64 or pentium 4. Shame I couldn't find any 17" screens on a decent cpu. :(