What you are interested in is called 'scale based visibility' which turns layers on and off depending on the zoom scale of the map. The QGIS help for it is here: https://docs.qgis.org/2.6/en/docs/training_manual/basic_map/symbology.html (section 3.2.5). With a little bit of work you can therefore construct a set of layers that automatically refreshes and changes display as you zoom in and out.
You can download mapping from OS Opendata which is intended to be displayed at various scales: https://www.ordnancesurvey.co.uk/opendatadownload/products.html. I think the order is (from big overview to local view):
- Meridian 2
You might only want some of that data, depending on your needs. You can have these in either raster or vector formats (vector allowing you to specify more about what features you want to display) and I think the raster is available in a muted background colourscheme. For large areas of raster data you should look at using virtual raster catalogs (https://docs.qgis.org/2.6/en/docs/training_manual/rasters/data_manipulation.html). For the vector data you can download style sheets here for the vector data: https://www.ordnancesurvey.co.uk/resources/carto-design/cartographic-stylesheets.html.
Add the data into the map and style them as required. Then set up the scale based visibility so that the maps are displayed at the correct zoom scale. You can then select all of them in the table of contents and do Save As -> Layer Definition File. Then, when you want your interactive mapping you can use Layer -> Add from layer definition file and your mapping will be added, with all the formatting and scale based visibility intact.
Not quite as easy as the basemaps available from ArcGIS but once you've set it up easy enough to use.
Obviously if you're not stuck on using OS mapping then you could check out Quick Map Services plugin - this allows you to add various online mapping services to QGIS but as far as I'm aware it doesn't support OS data at present.