iPhone 4S battery and WiFi issues

I have a dying battery (Genius diagnostic confirmed) and WiFi problems with poor connectivity/no networks found/one instance of greyed out fixed via the hair-drier method as software reset/reinstall didn't fix it.

2 questions.

Is it possible the battery is causing the WiFi issues ? I'm hoping a DIY battery replacement will fix the WiFi or is that just too optimistic.

If I need a new WiFi chip can anyone recommend a UK (ideally London) place to get the work done (Genius confirmed Apple don't fix this)

Thanks