It should be fine, so long as some of the work can be sorted out before the end of April.
You need to instruct a solicitor to do the legal work and your buyer and your vendor/seller need to instruct theirs. Then the Estate Agent will send a sales memorandum to all parties so that the work can begin. Your solicitor will wait for draft contract papers to be sent from your vendor's solicitors and at the same time your solicitor will send draft contract papers to your buyer's solicitor. Draft contract papers consist of a draft contract, draft transfer, title deeds and plans, property information form, fixtures & fittings, leasehold information form (leasehold only) as well as other documents relating to the property, i.e. EPC, Fensa, NHBC, building regulations and planning permissions etc.
You need to provide ID to your solicitor - usually passport/driving licence and a couple of bank statements. If you can't physically visit your solicitor, the ID will have to be verified by certain professionals and then posted/scanned/emailed.
Your solicitors will advise you what searches you might need (check with your solicitor if you have to pay separately from his fees for these).
You may wish to get an independent survey (i.e. not just rely on the mortgage lender's valuation survey)
Then there will be a gap while everyone waits for all the paperwork to come in and be distributed. By the time everything is to hand and the solicitors can raise enquiries over anything in the paperwork which needs clarifying, you should have your mortgage offer.