Kat-kin my advice would be to talk to your buyer yourself if you can, or use your estate agents to do it for you if you can't, and agree upon a Completion Date between you. Then you both inform your respective solicitors of this date and tell them you would like to Exchange Contracts, say one, two, or however many weeks suits you and your buyer, before the Completion Date. The minute you have Exchange of Contracts you can then both book your removal company with confidence. It is quite common nowadays to have Exchange and Completion on the same day but the trouble with that is it can all fall apart, even on the day, as no contracts have been exchanged. BTW it is a poor excuse from your buyer that their solicitor was away on holiday for 6 weeks as another solicitor should have been covering the work during that time. Your estate agents should be willing to help you set these dates as they don't get paid until Completion. The quickest I have ever moved was 6 weeks, the slowest was 4 months.
So agree your dates with buyer, inform solicitor and make sure these are achievable, once exchanged book removals. Good Luck.