Excellent advice here (one troll ... robertorolfo).
1. The bike2work scheme is good advice as it allows salary sacrifice to purchase bike equipment. Your employer may do it. I haven't used it yet as I just bought a cheap bike with cash. If you're in London there may be a subsidised TfL pass that you barter with HR for commuting expenses.
2. Stretching the budget is excellent advice. There are nice bikes from a British brand.
https://whyte.bike/collections/fast-...uter-r7-series
3. Fenders/Locks/Lights will eat that budget very quite quickly. When I moved to the UK, I bought one decent D-lock and two bright flashing USD-charing LEDs and fenders. That worked out to £100 from Amazon. Then fenders just broke after 500+ RT commutes. The lock and lights are still good.
About the budget...
When I moved to Frankfurt, I left my city bikes in Stockholm as I moved by train (yes, by rail, with the X2000 and ICE). I splurged and bought a CUBE hardtail MTB/helmet/lights/lock for around €900 or so.
I loved that bike and commuted every single day for 3 years on it. (Except when the bikepaths had ice which wasn't that often).
Thus, I would stretch the budget to £500 with all accessories including a backpack and a rainjacket etc... as you'll have equipment that you love every time that you use it as 11-miles one-way is a long trip for compromises.
Have fun and enjoy it!