At my surgery a 4 week wait is normal and that's with ANY GP. There are emergency slots but those are generally given to serious conditions or children and even getting someone to answer the phone in the morning is a challenge. They are also going to limit the timeframe you can call to make an appt from this month as well to 2 hours in the morning. Outwith this time, they will not book an appt!!
And no, it's not a large inner city practice either - they've just built lots of new houses and not recruited enough staff to cover the increase.
Had a blood test taken before Christmas but NO appts available to discuss the results. Been told they will phone "if there's something serious though" well, that's comforting..,