I think if you act quickly on this you could get it all sorted in 12 weeks no bother but theres many places it could fall down, and your position will be a complication - youll really need to pass as much of this off onto your partner as possible.
With regards getting your house sold quickly - decide whats the maximum you could get for the house (based on the recent sales in the area), then decide what you are willing to settle for in order to close a deal quickly. E.g. if the agent wants to guide at 300k in the hope of getting 330k (as is quite common in D12 right now) ask yourself if you would be willing to settle for 317.5k. You are guaranteed to get loads of bids at this price, allowing you to choose the best buyer (i.e. who has formal loan approval ready to go, and does not have a house to sell, has solicitor ready etc.).
You can then get the agent to tell the buyer that this is a time limited offer and if they do not sign contracts (in which you will add a clause specifying the latest closing date of 3 weeks time) within a week then you will cancel the sale and give it to another bidder. When I was looking for a house in D12 these are the exact kind of deals I was looking out for - sellers that needed to sell to the first reasonable offer and it worked out well for us (closed in 3.5 weeks for a price under the going rate for the area).
Thats your house sold, now focus on what you are buying: is it a new house? If so you should expect a delay in the date you actually get the keys - new builds are forever being delayed by a month, then another month etc. If its not a new house is it in walk in condition or does it need much work? Don't underestimate the hassle involved in moving all your stuff either. Are the sellers in a chain, as this could also be another sticking point?