Although it is technically possible, it is unrealistic to expect your cheque within 10 working days (still 2 weeks) unless you have all the necessary up to date documents to hand ie salary certificates, mortgage/bank/loan statements etc. You also need to have your life cover arranged and ammend your home insurance policy.
The 4 week time frame that Ulster Bank seems more realistic and they were probably even covering themselves for delays. Once you have all the documents required it shouldn't take any longer to arrange it with Ulster Bank than with EBS and you will be availing of a better rate with your legal fes still paid for.
I'm not saying that UB are 'better' than EBS, but you shouldn't choose the EBS simply because they are telling you that you will have your cheque within 10 days.