Whilst you may not see the avast tray icon, it isn’t part of the security side of avast, just the access to the program. On occasion my tray icon takes a little time to appear (not 60 seconds though), but if you open the task manager you will probably see it is running, avastUI.exe. You should also see the main avast service/function avastSvc.exe. If you catch it (start task manager) early enough you may also see some other avast functions avast.setup checking for virus signature updates (VPS).

Given that the times are broadly similar so there must be something else in the mix, which causes scan activity, other security based software, anti-spyware, Windows Defender, etc. ?

My spec is broadly similar to yours, but I used to have 2GB on this XP System and a couple of months ago I upgraded that to 4GB (the max for XP 32bit). Given this is an XP Pro system I would say it might be under resourced for all that you may have running on it, so there is likely to be a lot of swapping data in and out of memory to the hard disk pagefile.sys. If you can upgrade your RAM to 2gb or even 4gb that should improve overall system responsiveness.

What is the CPU that you have ?

Check the programs that you have installed and be strict on what you allow to run on boot (many don’t need to be), Operating System processes, Security based software and absolutely essential programs only. Boot is a busy time and the more programs you have loading the 1GB of RAM you have won’t be able to cope and had to swap out data in memory to the hard disk.