Since you have already put an offer on the house, not much use telling you what it is good to look for .
I would think in terms of mortgage and moving expenses: get an independent financial adviser town and country are good. They will find you a mortgage at a competitive deal and are paid by the lender so the advice is free to you (that certainly used to be the case, haven't used them since the credit crunch, heard another big brokers now charges for advice). I would suggest fixing now while interest is so low and I would definitely recommend you get a repayment mortgage rather than interest only (even with investment plan, it's very risky).
Your mortgage company will insist on some sort of valuation and it is probably worth getting a survey as well (you don't have to use your mortgage company's surveyor, it may be more cost effective to use an independent person for this but you will have to use their man for a valuation). Try not to get into a situation where you are tied into paying for your insurance through the mortgage lender and massive (and often virtually worthless) payment protection plans.
Get a good solicitor who you have a direct phone number for (rather than some company where you never quite know who is handling your case). Allow about £400-700 for conveyancing (including land registry fees etc), I don't know if you will be paying stamp duty but obviously factor that in. Think about how you are going to manage the move (removals company, man with van, family and friends?)
Does the house need work? Might it be worth staying where you are while this is carried out? If it does need something doing, then start getting builders in now to give you a quote for it.
If possible, get the vendor's contact numbers so that if there are any delays/unforeseen circumstances, you can speak to him direct rather than go through agent who will forget to pass message on or end up making whatever has happened sound much worse than it is. Try to keep the dialogue open (without harrassing them) because moving is really stressful and it is especially horrible if you don't know whether the other side is committed or not.
I'm sure other people will come up with lots of other stuff as well. Best of luck - average offer to completion cycle takes 13 weeks (perhaps longer now) so be patient.