The only times I’ve diverted from LGW when the runway was closed, the response from ATC went along the lines of.. LTN is full, STN and BHX will accept you if you declare an emergency, there is space at EMA or the next available airport is LPL... not sure if this is relevant to the OP but as has been said, with the flow rate into LGW, when the runway shuts, the surrounding airports fill up very quickly.