Unfortunately, not. Most household food waste is composted at anaerobic digester plants, which work without heat or oxygen and can’t break down solid packaging, which includes your compostable food bag. This means any packaging gets pulled out for waste to energy incineration.
Even if your food waste caddy goes to one of the 18 plants in the UK that accepts compostable plastics, they wouldn’t know it isn’t an oil-based plastic ( there isn’t a person vetting every non-food item), so it would get removed. The company Vegware are working on a scheme called the Composting Collective so large institutions and cafes can take back their packaging and pay to get it disposed of at their local in-vessel composter. There are also big institutions like universities sending their compostable packaging to these in-vessel composters too.