No problem. I think one of the experts should weigh in- but that setup is still not right. The vent needs to come AFTER the trap, not before. That sink vent is useless-- and it needs to go up and out of the house to allow sewer gases to exit the home. Otherwise, you will have a sewer gas problem- and sewer gases can make you very ill or even kill you if it builds up in large enough amounts.
Look at your drawing and tilt your head to the left. You see how the drain in black under the sink forms an S? It should go straight after the U part and then have a vent before going down. If the sink is a large utility sink, some people just secure the hose to the sink basin and let it drain. But it looks like you have a different configuration.
Here's a thread on another forum discussing the laundry & laundry tub plumbing
https://www.opia.info/index.php/code-tech-database/17-fixtures/79-13-25-standpipe-to-laundry-tub but that is Canadian code, I don't know the code in London.
But the drawing would have the sink with a 90 sweep instead of a trap and it would go to the standpipe and then have a trap.
So the recommendation is to tie the sink in to the standpipe rather than tie the laundry hose in to the sink. It is also recommended to have a cleanout on the vertical pipe with the vent (somewhere above the sanitary tee- probably best to be above the flood level rim).
Caveat: I am not an expert. I have just studied plumbing but am not an actual plumber. Frodo and the others would know more so if they disagree with me, listen to them (if they weigh in).
Keep in mind that the trap arm needs to be within certain parameters: too short and it siphons, too long and the air gets blocked, which interferes with water flow.
If you absolutely can't connect to the main vent pipe, then you might be able to get away with using an Air Admittance Valve (if it is allowed in your jurisdiction) but it is not recommended. In the drawing, you would put the AAV at the top of the vent pipe. I altered your drawing, but again, one of the experts should weigh in before this is finalized.